# Analytic Besov functional calculus for several commuting operators

Charles Batty St. John’s College, University of Oxford, Oxford OX1 3JP,
England<EMAIL_ADDRESS>,  Alexander Gomilko Faculty of Mathematics
and Computer Science
Nicolas Copernicus University
Chopin Street 12/18
87-100 Toruń, Poland<EMAIL_ADDRESS>,  Dominik Kobos USC School of
Philosophy
Mudd Hall of Philosophy (MHP)
3709 Trousdale Parkway
Los Angeles, CA 90089-0451
U.S.A<EMAIL_ADDRESS>and  Yuri Tomilov Institute of Mathematics
Polish Academy of Sciences
Śniadeckich 8
00-656 Warsaw, Poland
and
Faculty of Mathematics and Computer Science
Nicolas Copernicus University
Chopin Street 12/18
87-100 Toruń, Poland<EMAIL_ADDRESS>
###### Abstract.

This paper investigates when analytic Besov functions of $n$ variables act on
the generators of $n$ commuting $C_{0}$-semigroups on a Banach space. The
theory for $n=1$ has already been published, and the present paper uses a
different approach to that case as well as extending to the cases when $n\geq
2$. It also clarifies some spectral mapping properties and provides some
operator norm estimates.

###### Key words and phrases:

Analytic Besov functions, functional calculus, commuting semigroups

###### 2020 Mathematics Subject Classification:

Primary 47A20; Secondary 32A70 47B44 47D03

Dedicated with much gratitude to E. Brian Davies on the occasion of his 80th
birthday

##  1\. Introduction

Let $A$ be an (unbounded) operator on a complex Banach space $X$, and assume
that $-A$ is the generator of a bounded $C_{0}$-semigroup $(e^{-tA})_{t\geq
0}$ on $X$. The so-called Hille-Phillips functional calculus for $A$ assigns a
bounded operator $f(A)$ to each function $f$ in the Hille-Phillips (HP)
algebra $\mathcal{LM}$. If $A$ is sectorial of angle less than $\pi/2$, so the
semigroup is bounded and holomorphic on a sector in ${\mathbb{C}}_{+}$, then
there is a theory of a holomorphic calculus for $A$ which may be applied to
bounded holomorphic functions on a sector, but there is no guarantee that this
calculus assigns a bounded operator to all bounded holomorphic functions.
However in the sectorial case there is a bounded calculus for $A$ for all
functions in the Banach algebra $\mathcal{B}^{1}$ of analytic Besov functions
on ${\mathbb{C}}_{+}$, as shown originally in [30]. We have recently developed
a theory of a $\mathcal{B}^{1}$-calculus, which applies to many semigroup
generators, but not to all of them. There is a $\mathcal{B}^{1}$-calculus for
$A$ if $A$ is sectorial of angle less than $\pi/2$ or if $X$ is a Hilbert
space and the semigroup $(e^{-tA})_{t\geq 0}$ is bounded. The theory of the
$\mathcal{B}^{1}$-calculus is set out in detail in [6] and [7], but readers
may prefer to read the summary in [9] in order to understand the setting of
$\mathcal{B}^{1}$.

Some applications of the $\mathcal{B}^{1}$-calculus are given in [6] and [7],
and others have been found subsequently. For example, a new variant of the
Katznelson-Tzafriri theorem for the $\mathcal{B}^{1}$-calculus has been proved
in [10], provided that $A$ has a $\mathcal{B}^{1}$-calculus. Estimates
involving functions in $\mathcal{LM}$ can be improved by using the
$\mathcal{B}^{1}$-calculus norm, instead of the HP-norm; the difference is
considerable in some cases.

Functional calculus is often used to investigate the various notions of joint
spectrum for several commuting operators. Such work has had limited success,
and work on functional calculi for two or more generators of
$C_{0}$-semigroups has been sparse. The Hille-Phillips calculus for bounded
$C_{0}$-semigroups extends to several commuting semigroups in a
straightforward fashion. The holomorphic calculus for sectorial operators can
be applied in some cases. Some partial results for several commuting
holomorphic semigroups have been obtained in [4], [5], [22] and other papers,
and a multi-variable version of the Bochner-Phillips functional calculus was
set up in [25]. Some functional calculi for non-holomorphic functions have
been introduced for operators with real spectrum (for example, in [13]), and
versions for several operators have been studied in [2], [3] and [26]. The
$\mathcal{B}^{1}$-calculus does not apply to generators of bounded
$C_{0}$-groups, except in very special cases. Nevertheless Arveson’s theory of
representations of locally abelian groups can be applied to generators of
bounded $C_{0}$-groups (see [12, Chapter 8]) and to finitely many commuting
$C_{0}$-groups, in ways which help to develop the theory of the
$\mathcal{B}^{1}$-calculus. In addition, the book [1, Chapter 3] develops a
functional calculus for tuples of commuting generators of $C_{0}$-groups from
a different perspective and uses it for the study of commutators and other
applications to mathematical physics.

A two-variable version $\mathcal{B}^{2}$ of $\mathcal{B}^{1}$ and a theory of
the $\mathcal{B}^{2}$-calculus for two commuting operators have been covered
in the thesis [21], using the same methods as in [6] and [7]. In this paper we
extend the results to an arbitrary finite number of variables and commuting
operators. Moreover, we use techniques which are quite different from those in
[6] and [7]. We base them on a reproducing formula for analytic Besov
functions (see Corollary 3.3), which was first used in [6], and later used in
different ways in [7] and [8]. This formula provides a route to the essential
properties of the $\mathcal{B}^{n}$-calculus: mapping $\mathcal{B}^{n}$ into
$L(X)$, showing it is a homomorphism, and that it is unique (see Section 5).
This route is more direct, and clearer, than the arguments used in [6] and
[7].

We introduce the Banach algebras $\mathcal{B}^{n}$, and their elementary
decomposition into $2^{n}$ subalgebras, in Section 2. In the case $n=1$, the
two subalgebras were an ideal $\mathcal{B}^{1}_{0}$ and the one-dimensional
space of constant functions. In the general case we need more sophisticated
notation in our presentation of the decomposition, but the outcomes in this
paper are natural generalisations from the one-variable and two-variable
cases. In Section 3, we obtain a reproducing formula for the elementary
components of a given function $f\in\mathcal{B}^{n}$ (Corollary 3.3). In
Section 4, we define the $\mathcal{B}^{n}$-calculus initially as a function
from $\mathcal{B}^{n}$ to $L(X,X^{**})$, and then we prove in Section 5 that
it is a functional calculus. These proofs proceed in an unusual way, as did
the corresponding proofs in [6]. In Sections 6-8, we establish various further
properties of the family of $\mathcal{B}$-calculi, including compatibility
across operators, spectral inclusions, and operator norm estimates.

### Preliminaries

Notation: For $n\in{\mathbb{N}}$, we let $I_{n}:=\\{1,2,\dots,n\\}$ and
$\mathcal{P}_{n}$ be the power-set of $I_{n}$. We denote elements of
$\mathcal{P}_{n}$ by symbols such as $\Omega,\Psi$. For
$\Omega\in\mathcal{P}_{n}$, let $\Omega^{c}:=I_{n}\setminus\Omega$, and
$|\Omega|$ be the cardinality of $\Omega$.

We use the notation ${\mathbb{R}}_{+}$ for the interval $[0,\infty)$,
${\mathbb{C}}_{+}$ for the open right half-plane
$\\{\lambda\in{\mathbb{C}}:\operatorname{Re}\lambda>0\\}$ and
$\overline{\mathbb{C}}_{+}$ for the closed right half-plane. Real numbers may
typically be represented by $s,t,\alpha,\beta,\gamma$, and complex numbers by
$z,\lambda,\nu,\zeta$.

We use bold print to denote elements of ${\mathbb{R}}^{n}$ and
${\mathbb{C}}_{+}^{n}$; for example, $\mathbf{t}$ denotes
$(t_{1},\dots,t_{n})$ in ${\mathbb{R}}^{n}$, $\mathbf{z}$ denotes an $n$-tuple
$(z_{1},z_{2},\dots,z_{n})$ in ${\mathbb{C}}_{+}^{n}$, and similarly
${\bm{\lambda}}$ denotes an $n$-tuple
$(\lambda_{1},\lambda_{2}\dots,\lambda_{n})$.

Holomorphic functions: For a holomorphic function $f$ on
${\mathbb{C}}_{+}^{n}$, we let $D_{j}f$ be the partial derivative of $f$ with
respect to the $j$th coordinate variable $z_{j}$ where $j\in I_{n}$. For
$\Omega\in\mathcal{P}_{n}$, we let $\mathcal{D}_{\Omega}$ be the composition
of the derivatives $D_{j}$ for $j\in\Omega$ (once each). If
$\Omega=\emptyset$, then $\mathcal{D}_{\emptyset}f=f$. For $\Omega=I_{n}$, we
may write $\mathcal{D}_{n}$ instead of $\mathcal{D}_{I_{n}}$.

Let $f\in H^{\infty}({\mathbb{C}}_{+}^{n})$. We will make use of the following
properties:

  * •

For ${\bm{\alpha}}\in(0,\infty)^{n}$,
$\displaystyle\sup_{\operatorname{Re}\mathbf{z}={\bm{\alpha}}}|f(\mathbf{z})|=\sup_{\operatorname{Re}\mathbf{z}\geq{\bm{\alpha}}}|f(\mathbf{z})|$
(Maximum Principle).

  * •

For $\mathbf{z}\in{\mathbb{C}}_{+}^{n}$,
$\displaystyle|\mathcal{D}_{\Omega}f(\mathbf{z})|\leq\frac{\|f\|_{\infty}}{2^{|\Omega|}\prod_{j\in\Omega}\operatorname{Re}z_{j}}$
(Cauchy’s inequality).

Cauchy’s inequality extends to higher-order derivatives, by repeated use of
first-order cases.

Several variants of resolvent functions appear frequently, including the
following:

  * •

For $z,\lambda\in{\mathbb{C}}_{+}$, we put $r_{\lambda}(z)=(z+\lambda)^{-1}$.

  * •

For $\lambda\in{\mathbb{C}}_{+}$, $j\in I_{n}$ and
$\mathbf{z}\in{\mathbb{C}}_{+}^{n}$, we put
$r_{\lambda,j}(\mathbf{z})=(z_{j}+\lambda)^{-1}$.

  * •

For $\mathbf{z},{\bm{\lambda}}\in{\mathbb{C}}_{+}^{n}$, we put
$\mathbf{r}_{\bm{\lambda}}(\mathbf{z})=\prod_{j=1}^{n}(z_{j}+\lambda_{j})^{-1}$.

  * •

For $z,\lambda\in{\mathbb{C}}_{+}$, we put
$K(z,\lambda)=(-2/\pi)(z+\lambda)^{-2}$.

  * •

For $\mathbf{z},{\bm{\lambda}}\in{\mathbb{C}}_{+}^{n}$, we put
${\mathcal{K}}_{n}(\mathbf{z},{\bm{\lambda}})=\prod_{j=1}^{n}K(z_{j},\lambda_{j})$.

Measures: We may denote Lebesgue measure on ${\mathbb{R}}_{+}^{n}$ by
$d{\bm{\alpha}}$ or $d{\bm{t}}$, and Lebesgue measure on
${\mathbb{C}}_{+}^{n}$ by $dS_{n}$. We let

$dV(\lambda)=\alpha\,d\beta\,d\alpha,\qquad\lambda=\alpha+i\beta\in{\mathbb{C}}_{+},$

and $dV_{n}$ be the $n$-fold product measure of $dV$, so

$dV_{n}({\bm{\lambda}})=\left(\textstyle{\prod}_{j=1}^{n}\alpha_{j}\right)\,dS_{n}({\bm{\lambda}}),\qquad{\bm{\lambda}}=(\alpha_{j}+i\beta_{j})_{j=1}^{n}\in{\mathbb{C}}_{+}^{n}.$

Operators: In this paper an “operator” may be either a bounded linear operator
on a Banach space $X$ or an unbounded operator with dense domain in $X$. The
notation $L(X)$ and $L(X,Y)$ denote the spaces of bounded linear operators
from $X$ to $X$ and from $X$ to $Y$, respectively. Bounded operators will
typically be denoted by $T$ and unbounded operators by $A$.

An operator $A$ on $X$ is sectorial of angle less than $\pi/2$ if and only if
the spectrum $\sigma(A)$ of $A$ is contained in ${\mathbb{C}}_{+}\cup\\{0\\}$
and
$M_{A}:=\sup_{\lambda\in{\mathbb{C}}_{+}}\|\lambda(\lambda+A)^{-1}\|<\infty$.
Equivalently, the operator $-A$ is the generator of a sectorially bounded
holomorphic $C_{0}$-semigroup on $X$.

##  2\. The algebra $\mathcal{B}^{n}$

Let $f$ be a holomorphic function on ${\mathbb{C}}_{+}^{n}$,
$\Omega\in\mathcal{P}_{n}$ and $k=|\Omega|$. Let $H_{\Omega}f$ be the function
of variables
${\bm{\alpha}}_{\Omega}:=(\alpha_{j})_{j\in\Omega}\in(0,\infty)^{k}$ for
$j\in\Omega$, defined by

(2.1) $(H_{\Omega}f)({\bm{\alpha}}_{\Omega}):=\sup_{\mathbf{z}\in
W_{{\bm{\alpha}}_{\Omega}}}\left|(\mathcal{D}_{\Omega}f)(\mathbf{z})\right|,$

where

$W_{{\bm{\alpha}}_{\Omega}}:=\left\\{\mathbf{z}\in{\mathbb{C}}_{+}^{n}:\text{$\operatorname{Re}z_{j}=\alpha_{j}$
for all $j\in\Omega$}\right\\}.$

If $f$ is bounded and $\alpha_{j}>0$ for each $j\in\Omega$, then Cauchy’s
inequality and the maximum principle imply that the supremum in (2.1) is
finite, and $H_{\Omega}f$ is a decreasing function of each variable
$\alpha_{j}$ for $j\in\Omega$.

We define

(2.2)
$\|f\|_{\mathcal{B}^{n}_{\Omega}}:=\int_{{\mathbb{R}}_{+}^{k}}(H_{\Omega}f)({\bm{\alpha}}_{\Omega})\,d{\bm{\alpha}}_{\Omega},$

where $d{\bm{\alpha}}_{\Omega}$ denotes Lebesgue measure on
${\mathbb{R}}_{+}^{k}$. This integral may be infinite. When $\Omega$ is the
empty set, $H_{\emptyset}f$ and $\|f\|_{\mathcal{B}^{n}_{\emptyset}}$ are both
equal to the $H^{\infty}$-norm $\|f\|_{\infty}$ of $f$.

We define $\mathcal{B}^{n}$ to be the vector space of all holomorphic
functions $f$ on ${\mathbb{C}}_{+}^{n}$ such that
$\|f\|_{\mathcal{B}^{n}_{\Omega}}$ is finite for all
$\Omega\in\mathcal{P}_{n}$, so that $H_{\Omega}f$ is integrable over
${\mathbb{R}}_{+}^{k}$. Then $\|\cdot\|_{\mathcal{B}^{n}_{\Omega}}$ is a
seminorm on $\mathcal{B}^{n}$, and there is a norm on $\mathcal{B}^{n}$
defined by

(2.3)
$\|f\|_{\mathcal{B}^{n}}:=\sum_{\Omega\in\mathcal{P}_{n}}\|f\|_{\mathcal{B}_{\Omega}^{n}}.$

For $n=1$, this definition agrees with the norm on $\mathcal{B}^{1}$ defined
in [6, p.33], and, for $n=2$, it agrees with the norm on $\mathcal{B}^{2}$
defined in [21, p.37].

The definitions of $\mathcal{B}^{n}$ and its norm are clearly invariant under
permutation of the variables. For ease of presentation, in a proof of any
statement involving one set $\Omega\in\mathcal{P}_{n}$, we may choose to
assume that $\Omega=I_{k}$ or $\Omega^{c}=I_{k}$.

A useful property of functions $f\in\mathcal{B}^{n}$ is that they are bounded
and uniformly continuous on ${\mathbb{C}}_{+}^{n}$, so they extend to bounded,
uniformly continuous functions on $\overline{{\mathbb{C}}}_{+}^{n}$. This was
noted for $n=1$ in [6, Proposition 2.1] and shown for $n=2$ in [21,
Proposition 3.2.4]. In general, the uniform continuity can be proved by some
epsilontics or by a special case of an argument in the proof of Remark 3.7(1).
As a preliminary taste, we will show here that both properties can be deduced
using only the finiteness of the norms $\|f\|_{\mathcal{B}^{n}_{\Omega}}$ for
every singleton subset $\Omega$ of $I_{n}$. However this is an exceptional
case, and we need to include the $H^{\infty}$-norm in the definition of
$\|f\|_{\mathcal{B}^{n}}$ in order that Proposition 2.2 is true and later
proofs can be simplified.

######  Proposition 2.1.

Let $f$ be a holomorphic function on ${\mathbb{C}}_{+}^{n}$, and assume that
$\|f\|_{\mathcal{B}^{n}_{\Omega}}$ is finite for every singleton subset
$\Omega$ of $I_{n}$. Then $f$ is bounded and uniformly continuous on
${\mathbb{C}}_{+}^{n}$.

###### Proof.

Let $\mathbf{z}$ and $\mathbf{z}^{\prime}$ be points in
${\mathbb{C}}_{+}^{n}$. To estimate $|f(\mathbf{z})-f(\mathbf{z}^{\prime})|$,
we will consider the variables one at a time. The first step will be from
$\mathbf{z}$ to $\mathbf{z}^{[1]}=(z_{1}^{\prime},z_{2},\dots,z_{n})$, as
follows.

Let $\theta\in(0,\pi/2)$ and $\kappa=\sec\theta>1$. Consider the following
path $\Gamma_{1}$ in ${\mathbb{C}}_{+}$ from $z_{1}$ to $z_{1}^{\prime}$:

  1. (a)

the line-segment from $z_{1}$ to $z_{1}^{\prime}$ if the gradient is in
$[-\cot\theta,\cot\theta]$,

  2. (b)

otherwise, two line-segments, one from $z_{1}$ and the other from
$z_{1}^{\prime}$, one with gradient $\cot\theta$ and the other with gradient
$-\theta$, chosen so that the two lines cross at a point
$z_{1}^{\prime\prime}$ to the right of $z$ and $z_{1}^{\prime}$.

Then

$|f(\mathbf{z})-f(\mathbf{z}^{[1]})|=\left|\int_{\Gamma_{1}}(D_{1}f)(\lambda,z_{2},\dots,z_{n})\,d\lambda\right|\leq\kappa\int_{J_{1}}H_{\\{1\\}}(t)\,dt.$

Here $J_{1}$ represents the interval in ${\mathbb{R}}_{+}$ between
$\operatorname{Re}z_{1}$ and $\operatorname{Re}z_{1}^{\prime}$ in case (a). In
case (b), it represents both the interval between $\operatorname{Re}z_{1}$ and
$\operatorname{Re}z_{1}^{\prime\prime}$ and the interval between
$\operatorname{Re}z_{1}^{\prime}$ and $\operatorname{Re}z_{1}^{\prime\prime}$,
with the two integrals added together. There is a similar estimate for
$|f(\mathbf{z}^{[1]})-f(\mathbf{z}^{[2]})|$, where
$\mathbf{z}^{[2]}=(z_{1}^{\prime},z_{2}^{\prime},z_{3},\dots,z_{n})$, and so
on. Eventually we obtain

$|f(\mathbf{z})-f(\mathbf{z}^{\prime})|\leq\kappa\sum_{j=1}^{n}\int_{J_{j}}H_{\\{j\\}}(t)\,dt\leq
2\kappa\sum_{j=1}^{n}\|f\|_{\mathcal{B}^{n}_{\\{j\\}}}<\infty.$

By fixing $\mathbf{z}^{\prime}$, this proves that $f$ is bounded. By letting
$\theta\to 0$, one may conclude that the estimate also holds for $\kappa=1$.

To establish uniform continuity, we fix $\theta\in(0,\pi/2)$ and thereby we
fix $\kappa>1$. Take $\delta>0$ and assume that
$|\mathbf{z}-\mathbf{z}^{\prime}|<\delta$. Then the length of each interval
involved in the sets $J_{j}$ is at most $\delta\cot\theta$. Since the
functions $H_{\\{\j\\}}$ are integrable over ${\mathbb{R}}_{+}$, there exists
$\delta>0$ so small that all the integrals used above are less than
$\varepsilon$, and then we can conclude that
$|f(\mathbf{z})-f(\mathbf{z}^{\prime})|<2\kappa n\varepsilon$ whenever
$|\mathbf{z}-\mathbf{z}^{\prime}|<\delta$. ∎

We choose the norm $\|\cdot\|_{\mathcal{B}^{n}}$ because it makes
$\mathcal{B}^{n}$ into a Banach algebra. There are numerous norms on
$\mathcal{B}^{n}$ which are equivalent to $\|\cdot\|_{\mathcal{B}^{n}}$.
Examples for $n=1$ may be found in [6, p.33] and for $n=2$ in [21, Section
3.4].

######  Proposition 2.2.

The normed space $(\mathcal{B}^{n},\|\cdot\|_{\mathcal{B}^{n}})$ is a Banach
algebra.

###### Proof.

The proof of completeness of the space is a straightforward, but
uninteresting, extension of the proofs for $\mathcal{B}^{1}$ in [6,
Proposition 2.3] and $\mathcal{B}^{2}$ in [21, Proposition 3.2.1].

To show that $\mathcal{B}^{n}$ is a Banach algebra, we need to show that, for
any $f,g\in\mathcal{B}^{n}$, the product $fg$ satisfies
$\|fg\|_{\mathcal{B}^{n}}\leq\|f\|_{\mathcal{B}^{n}}\|g\|_{\mathcal{B}^{n}}$.

Let $\Omega\in\mathcal{P}_{n}$. By the product rule,

$\mathcal{D}_{\Omega}(fg)=\sum_{\Psi\subseteq\Omega}(\mathcal{D}_{\Psi}f)(\mathcal{D}_{\Omega\setminus\Psi}g).$

For $\Psi\subseteq\Omega$,

$\displaystyle\hbox{}\hskip-30.0pt\sup\\{|(\mathcal{D}_{\Psi}f)(\mathcal{D}_{\Omega\setminus\Psi}g)(\mathbf{z})|:\mathbf{z}\in{\mathbb{C}}_{+}^{n},\operatorname{Re}z_{j}=\alpha_{j}\,(j\in\Omega)\\}\hskip
20.0pt$ $\displaystyle\hbox{}\hskip
15.0pt\leq\sup\\{|(\mathcal{D}_{\Psi}f)(\mathbf{z}))|:\mathbf{z}\in{\mathbb{C}}_{+}^{n},\operatorname{Re}z_{j}=\alpha_{j}\,(j\in\Psi)\\}$
$\displaystyle\hbox{}\hskip
30.0pt\times\sup\\{|(\mathcal{D}_{\Omega\setminus\Psi}g)(\mathbf{z})|:\mathbf{z}\in{\mathbb{C}}_{+}^{n},\operatorname{Re}z_{j}=\alpha_{j}\,(j\in\Omega\setminus\Psi)\\}.$

The variables $\alpha_{j}$ for $j\in\Psi$ separate from those for
$j\in\Omega\setminus\Psi$ on the right-hand side, so integration with respect
to Lebesgue measure on ${\mathbb{R}}_{+}^{n}$ shows that

$\|fg\|_{\mathcal{B}^{n}_{\Omega}}\leq\sum_{\Psi\subseteq\Omega}\|f\|_{\mathcal{B}^{n}_{\Psi}}\|g\|_{\mathcal{B}^{n}_{\Omega\setminus\Psi}}.$

Hence

$\displaystyle\|fg\|_{\mathcal{B}^{n}}\leq\sum_{\Omega\in\mathcal{P}_{n}}\sum_{\Psi\subseteq\Omega}\|f\|_{\mathcal{B}^{n}_{\Psi}}\|g\|_{\mathcal{B}^{n}_{\Omega\setminus\Psi}}\leq\sum_{\Psi\in\mathcal{P}_{n}}\|f\|_{\mathcal{B}^{n}_{\Psi}}\sum_{\Psi\in\mathcal{P}_{n}}\|g\|_{\mathcal{B}^{n}_{\Psi}}=\|f\|_{\mathcal{B}^{n}}\|g\|_{\mathcal{B}^{n}}.$

∎

We note the following facts, which extend the corresponding statements for
$n=1$ or $n=2$. The first result is almost immediate from the definitions. The
other proofs are either similar to those in [6] and [21], or, in some cases,
they can be deduced from those results.

######  Proposition 2.3.

Let $f\in\mathcal{B}^{n}$, $\Omega\in\mathcal{P}_{n}$ with $|\Omega|=k$, and
$\zeta_{j}\in{\mathbb{C}}_{+}$ be fixed for $j\in\Omega^{c}$. For
$\mathbf{z}\in{\mathbb{C}}_{+}^{n}$, let
$\mathbf{z}_{\Omega}=(z_{j})_{j\in\Omega}$, and
$\mathbf{z}_{\Omega}^{+}=(z^{*}_{j})_{j\in I_{n}}$, where

$z^{*}_{j}=\begin{cases}z_{j},&j\in\Omega,\\\
\zeta_{j},&j\in\Omega^{c}.\end{cases}$

Let $\Psi$ be a subset of $\Omega^{c}$, and

$g(\mathbf{z}_{\Omega})=(\mathcal{D}_{\Psi}f)(\mathbf{z}_{\Omega}^{+}).$

Then $g\in\mathcal{B}^{k}$. In particular, the function
$\mathbf{z}_{\Omega}\mapsto f(\mathbf{z}_{\Omega}^{+})$ is in
$\mathcal{B}^{k}$.

###### Proof.

It suffices to consider the case when $\Omega=I_{n-1}$ and show that
$g\in\mathcal{B}^{n-1}$, in the cases when $\Psi$ is empty or $\Psi=\\{n\\}$.
Then, we can conclude the result for general $\Omega$ by the symmetry of the
variables, and considering the variables in $\Omega^{c}$ successively.

Let $\Upsilon\in\mathcal{P}_{n-1}$. If $\Psi$ is empty, then it is clear that
$(H_{\Upsilon}g)({\bm{\alpha}}_{\Upsilon})\leq(H_{\Upsilon}f)({\bm{\alpha}}_{\Upsilon})$,
and so $g\in\mathcal{B}^{n-1}$. In the case when $\Psi=\\{n\\}$, we consider
$\zeta_{n}\in{\mathbb{C}}_{+}$, $\Upsilon\in\mathcal{P}_{n-1}$ with
$|\Upsilon|=m$ and $\Upsilon^{+}=\Upsilon\cup\\{n\\}$. Since
$f\in\mathcal{B}^{n}$, the function

$({\bm{\alpha}}_{\Upsilon},\alpha_{n})\mapsto(H_{\Upsilon^{+}}f)(\alpha_{\Upsilon},\alpha_{n})$

is integrable over ${\mathbb{R}}_{+}^{m+1}$. By Fubini’s theorem, the function

${\bm{\alpha}}_{\Upsilon}\mapsto(H_{\Upsilon^{+}}f)(\alpha_{\Upsilon},\alpha_{n})$

is integrable over ${\mathbb{R}}_{+}^{m}$ for almost all
$\alpha_{n}\in(0,\infty)$. By the maximum principle, these functions of
${\bm{\alpha}}_{\Upsilon}$ form a family of non-negative functions which are
non-increasing with respect to $\alpha_{n}$, and the functions are integrable
over ${\mathbb{R}}_{+}^{m}$ for almost all $\alpha_{n}>0$, so they are
integrable for all $\alpha_{n}>0$. Moreover,

$(H_{\Upsilon}g)({\bm{\alpha}}_{\Upsilon})\leq(H_{\Upsilon^{+}}f)({\bm{\alpha}}_{\Upsilon},\operatorname{Re}\zeta_{n}),$

so $H_{\Upsilon}g$ is integrable. ∎

Let $f\in\mathcal{B}^{n}$, and $\Omega\in\mathcal{P}_{n}$. The following
result was shown in [6, Proposition 2.2] for $n=1$, and in [21, Proposition
3.2.2] for $n=2$. The existence of iterated limits follows by applying the
one-variable result repeatedly, but some additional arguments are needed to
obtain a limit over several variables simultaneously.

######  Proposition 2.4.

Let $f\in\mathcal{B}^{n}$, $\Omega\in\mathcal{P}_{n}$, $|\Omega|=k$, and
$\mathbf{z}_{\Omega}=(z_{j})_{j\in\Omega}\in{\mathbb{C}}_{+}^{k}$. Then the
following limit exists in ${\mathbb{C}}$:

$f_{\Omega}(\mathbf{z}_{\Omega}):=\lim_{\operatorname{Re}z_{j}\to\infty,j\in\Omega^{c}}f(\mathbf{z}).$

Moreover, $f_{\Omega}\in\mathcal{B}^{k}$, and the map $f\mapsto f_{\Omega}$ is
bounded and linear from $\mathcal{B}^{n}$ to $\mathcal{B}^{k}$.

###### Proof.

By symmetry of the variables, it suffices to consider the case when
$\Omega=I_{k}$. First we show the result when $k=n-1$. For this case, we use
the notation $\mathbf{z}$ to denote
$(z_{1},\dots,z_{n-1})\in{\mathbb{C}}_{+}^{n-1}$, and $\zeta$ for the variable
$z_{n}$.

By Proposition 2.3, for each $\mathbf{z}\in{\mathbb{C}}_{+}^{n-1}$ the
function $\zeta\mapsto f(\mathbf{z},\zeta)$ is in $\mathcal{B}^{1}$. By [6,
Proposition 2.2], we know that

$g_{1}(\mathbf{z}):=\lim_{\operatorname{Re}\zeta\to\infty}f(\mathbf{z},\zeta)$

exists. In fact,

(2.4) $\displaystyle\left|f(\mathbf{z},\zeta)-g_{1}(\mathbf{z})\right|$
$\displaystyle\leq\left|\int_{\operatorname{Re}\zeta}^{\infty}(D_{n}f)(\mathbf{z},t+i\operatorname{Im}\zeta)\,dt\right|\leq\int_{\operatorname{Re}\zeta}^{\infty}(H_{\\{n\\}}f)(t)\,dt\to
0,$

uniformly with respect to $\mathbf{z}\in{\mathbb{C}}_{+}^{n-1}$, as
$\operatorname{Re}\zeta\to\infty$. It follows from Vitali’s theorem for
several variables that $g_{1}$ is holomorphic, and the partial derivatives of
$f(\cdot,\zeta)$ converge (pointwise) to the corresponding derivatives of
$g_{1}$ as $\operatorname{Re}\zeta\to\infty$. This implies that
$H_{\Upsilon}g_{1}\leq H_{\Upsilon}f$ on $(0,\infty)^{|\Upsilon|}$ for all
$\Upsilon\in\mathcal{P}_{n-1}$, and therefore $g_{1}\in\mathcal{B}^{n-1}$ and
$\|g_{1}\|_{\mathcal{B}^{n-1}}\leq\|f\|_{\mathcal{B}^{n}}$.

If $\Omega=I_{k}$ where $k\in\\{0,1,\dots,n-2\\}$, one may apply the argument
above to $g_{1}$ to show that there is a function $g_{2}\in\mathcal{B}^{n-2}$
such that

$\lim_{\operatorname{Re}\zeta\to\infty}\big{(}g_{1}(\mathbf{z}^{\prime},\zeta)-g_{2}(\mathbf{z}^{\prime})\big{)}=0,$

uniformly for $\mathbf{z}^{\prime}\in{\mathbb{C}}_{+}^{n-2}$. In combination
with (2.4), it follows that the statement is proved for $\Omega=I_{n-2}$.
Continuing in this way, it can be proved for $\Omega=I_{k}$. ∎

######  Remarks 2.5.

1\. In Proposition 2.4, $f_{I_{n}}=f$ and $f_{\emptyset}$ is the constant
function $\lim_{\operatorname{Re}z_{j}\to\infty,\,j\in I_{n}}f(\mathbf{z})$.

2\. Note that the limits involved in Proposition 2.4 are as
$\operatorname{Re}z_{j}\to\infty$. Functions $f\in\mathcal{B}^{n}$ may not
have limits as $|z_{j}|\to\infty$.

3\. The function $f_{\Omega}$ in Proposition 2.4 may be denoted in various
ways. In particular, we may replace the variable $z_{j}$ by the symbol
$\infty$ for each $j\in\Omega^{c}$, and then regard $f$ as being defined on an
extended domain. Proposition 2.3 remains true if $\zeta_{j}=\infty$ for some
$j\in\Omega^{c}$. The function $f_{\Omega}$ is holomorphic, by a simple
application of Vitali’s theorem, and then the rest of the proof remains valid.

We denote by $\mathcal{B}_{0}^{n}$ the ideal of $\mathcal{B}^{n}$ consisting
of those functions $f\in\mathcal{B}^{n}$ such that $f_{\Omega}$ is identically
zero for all proper subsets $\Omega$ of $I_{n}$. It suffices that

$\lim_{\operatorname{Re}z_{j}\to\infty}f(\mathbf{z})=0$

whenever $j\in I_{n}$ and $z_{k}\in{\mathbb{C}}_{+}$ are fixed for each $j\neq
k$. Adopting the practice described in Remark 2.5, it suffices that
$f\in\mathcal{B}_{0}^{n}$ if and only if $f(\mathbf{z})=0$ whenever
$z_{j}=\infty$ for at least one $j\in I_{n}$. We stress that the two sentences
above mean precisely the same.

The seminorm $\|\cdot\|_{\mathcal{B}^{n}_{I_{n}}}$ is a norm on
$\mathcal{B}_{0}^{n}$, and we denote this seminorm by
$\|\cdot\|_{\mathcal{B}_{0}^{n}}$. Now we show that the norms
$\|\cdot\|_{\mathcal{B}_{0}^{n}}$ and $\|\cdot\|_{\mathcal{B}^{n}}$ are
equivalent on $\mathcal{B}^{n}_{0}$. In addition, finiteness of
$\|f\|_{\mathcal{B}^{n}_{0}}$, boundedness of $f$, and the condition above of
vanishing (at infinity, to the right) imply that $f\in\mathcal{B}^{n}_{0}$.

######  Proposition 2.6.

If $f\in H^{\infty}({\mathbb{C}}_{+}^{n})$,
$\|f\|_{\mathcal{B}^{n}_{0}}<\infty$ and assume that

(2.5) $\lim_{{\rm Re}z_{j}\to\infty}f(\mathbf{z})=0$

whenever $j\in I_{n}$ and $z_{k}$ is fixed for all $k\neq j$. Then
$f\in\mathcal{B}^{n}$ and $\|f\|_{\mathcal{B}^{n}}\leq
2^{n}\|f\|_{\mathcal{B}_{0}^{n}}$.

###### Proof.

Let $\Omega\in\mathcal{P}_{n}$, $\Omega\not=I_{n}$, and $|\Omega|=k$. From the
boundedness of $f$, (2.5) and Cauchy’s inequality, we infer that for all
$\Psi\in\mathcal{P}_{n}$ such that $\Omega\subseteq\Psi$, any $j\in\Omega^{c}$
and $z_{k}$ fixed for $k\neq j$,

$\lim_{{\rm Re}\,z_{j}\to\infty}\,\mathcal{D}_{\Psi}f(\mathbf{z})=0,$

and thus

$\mathcal{D}_{\Omega}f(\mathbf{z})=\int_{{\mathbb{R}}_{+}^{n-k}}\mathcal{D}_{n}f(\mathbf{z}+{\bm{\alpha}}_{\Omega^{c}})\,d{\bm{\alpha}}_{\Omega^{c}},\qquad\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

By the maximum principle for functions from $H^{\infty}(\mathbb{C}^{n})$, we
have

$\displaystyle|\mathcal{D}_{n}f(\mathbf{z})|$
$\displaystyle\leq\int_{{\mathbb{R}}_{+}^{n-k}}\sup_{\begin{subarray}{c}\lambda_{j}=z_{j},\,j\in\Omega\\\
\operatorname{Re}\lambda_{j}=\operatorname{Re}z_{j},\,j\in\Omega^{c}\end{subarray}}|\mathcal{D}_{n}f({\bm{\lambda}}+{\bm{\alpha}}_{\Omega^{c}})|\,d{\bm{\alpha}}_{\Omega^{c}}$
$\displaystyle\leq\int_{{\mathbb{R}}_{+}^{n-k}}\sup_{\begin{subarray}{c}\lambda_{j}=z_{j},\,j\in\Omega\\\
\operatorname{Re}\lambda_{j}=0,\,j\in\Omega^{c}\end{subarray}}|\mathcal{D}_{n}f({\bm{\lambda}}+{\bm{\alpha}}_{\Omega^{c}})|\,d{\bm{\alpha}}_{\Omega^{c}},$

so that

$\|f\|_{\mathcal{B}_{\Omega}^{n}}=\sup_{\mathbf{z}\in
W_{{\bm{\alpha}}_{\Omega}}}\,|\mathcal{D}_{\Omega}f(\mathbf{z})|\leq\int_{{\mathbb{R}}^{n-k}_{+}}\,\sup_{\mathbf{z}\in
W_{{\bm{\alpha}}_{I_{n}}}}|\mathcal{D}_{n}f(\mathbf{z})|\,d{\bm{\alpha}}_{\Omega^{c}}=\|f\|_{\mathcal{B}^{n}_{0}}.$

This implies that $f\in\mathcal{B}_{0}^{n}$ and

$\|f\|_{\mathcal{B}^{n}}=\sum_{\Omega\in\mathcal{P}_{n}}\|f\|_{\mathcal{B}^{n}_{\Omega}}\leq
2^{n}\|f\|_{\mathcal{B}_{0}^{n}}.\qed$

Note that $\|\cdot\|_{\mathcal{B}^{n}_{I_{n}}}$ is not a norm on
$\mathcal{B}^{n}$, because some functions do not depend on all the $n$
variables. Those functions satisfy $\mathcal{D}_{n}f=0$ and hence
$\|f\|_{\mathcal{B}^{n}_{I_{n}}}=0$. However linear combinations $g$ of such
functions may depend on all the variables while satisfying
$\mathcal{D}_{n}g=0$ (see Remark 2.9(2) below). Consequently we will decompose
$\mathcal{B}^{n}$ into the direct sum of $2^{n}$ closed ideals, one for each
subset of $I_{n}$. This was achieved for $\mathcal{B}^{1}$ in [6], where one
summand contained only the constant functions, and also for $\mathcal{B}^{2}$
in [21], where two summands each contained functions of one of the two
variables.

Let $f\in\mathcal{B}^{n}$, and let $\Omega_{f}\in\mathcal{P}_{n}$ be the
subset of $I_{n}$ representing the variables $z_{j}$ on which $f$ depends. It
is easy to see that they are the variables for which $D_{j}f$ is not
identically zero. We will use the following terminology:

  1. (a)

$\Omega_{f}$ is the support of $f$,

  2. (b)

$|\Omega_{f}|$ is the degree of $f$,

  3. (c)

$f$ is elementary if $f(\mathbf{z})=0$ whenever $z_{j}=\infty$ for at least
one $j\in\Omega_{f}$.

For $\Omega\in\mathcal{P}_{n}$, we define subspaces of $\mathcal{B}^{n}$:

$\displaystyle\mathcal{B}^{n}_{\Omega}$
$\displaystyle=\left\\{f\in\mathcal{B}^{n}:\Omega_{f}\subseteq\Omega\right\\}=\left\\{f\in\mathcal{B}^{n}:\text{$\mathcal{D}_{j}f=0$
for all $j\in\Omega^{c}$}\right\\},$ $\displaystyle\mathcal{B}^{n}_{\Omega,0}$
$\displaystyle=\left\\{f\in\mathcal{B}^{n}_{\Omega}:\text{$f(\mathbf{z})=0$
whenever $z_{j}=\infty$ for at least one $j\in\Omega$}\right\\}.$

It is easy to see that $\mathcal{B}^{n}_{\Omega}$ and
$\mathcal{B}^{n}_{\Omega,0}$ are closed subalgebras of $\mathcal{B}^{n}$. The
spaces $\mathcal{B}^{n}_{I_{n}}$ and $\mathcal{B}^{n}_{I_{n},0}$ are the
spaces $\mathcal{B}^{n}$ and $\mathcal{B}^{n}_{0}$, respectively, and the
space $\mathcal{B}^{n}_{\emptyset}$ is the space of constant functions. Note
that

  1. (i)

$f\in\mathcal{B}^{n}$ is elementary if and only if
$f\in\mathcal{B}^{n}_{\Omega_{f},0}$, and

  2. (ii)

If $f\in\mathcal{B}^{n}_{0}$ and $\mathcal{D}_{n}f=0$, then $f=0$.

For $f\in\mathcal{B}^{n}$, let

$f_{\rm
el}(\mathbf{z})=\sum_{\Omega\in\mathcal{P}_{n}}(-1)^{n-|\Omega|}f_{\Omega}(\mathbf{z}_{\Omega}),$

where $f_{\Omega}(\mathbf{z}_{\Omega})$ is defined in Proposition 2.4. For
example, if $n=2$ then

$f_{\rm
el}(z_{1},z_{2})=f(z_{1},z_{2})-f(z_{1},\infty)-f(\infty,z_{2})+f(\infty,\infty).$

Then

$f=f_{\rm el}+\sum_{\Omega\in\mathcal{P}_{n},\Omega\neq
I_{n}}(-1)^{n-|\Omega|-1}f_{\Omega},$

where $f_{\rm el}$ is elementary of degree $n$, and each function $f_{\Omega}$
for $\Omega\neq I_{n}$ is of degree less than $n$. The empty set is in
$\mathcal{P}_{n}$, and $f_{\emptyset}=f_{\emptyset,0}$ is a constant equal to
$f(\infty,\dots,\infty)$.

By simple relabelling of the variables, the space $\mathcal{B}_{\Omega}^{n}$
can be identified with $\mathcal{B}^{k}$ and $\mathcal{B}^{n}_{\Omega,0}$ with
$\mathcal{B}^{k}_{0}$, where $k=|\Omega|$. By considering those $f_{\Omega}$
for $\Omega\neq I_{n}$, and iterating the procedure until we have obtained
only elementary functions, we can obtain an elementary decomposition of $f$ as

(2.6) $f=\sum_{\Omega\in\mathcal{P}_{n}}f_{\Omega,0},$

where $f_{\Omega,0}$ is either identically zero or an elementary function with
support $\Omega$.

The elementary decomposition of $f\in\mathcal{B}^{n}$ is unique. Suppose that
$\sum_{\Omega\in\mathcal{P}_{n}}f_{\Omega}=0$, where $f_{\Omega}$ is
elementary with support $\Omega$. By setting $z_{j}=\infty$ for all $j\in
I_{n}$, one obtains that the constant function $f_{\emptyset}=0$. By setting
$z_{j}=\infty$ for all $z_{j}$ except for $j=1$, one obtains that
$f_{\\{1\\}}=0$. Continuing in this way one obtains that $f_{\Omega}=0$ for
all $\Omega$.

So far the discussion has been somewhat heuristic. An alternative description
is given in the following formulas, which can be readily verified.

######  Proposition 2.7.

The Banach algebra $\mathcal{B}^{n}$ is the direct sum of the closed
subalgebras $\mathcal{B}^{n}_{\Omega,0}$ for $\Omega\in\mathcal{P}_{n}$. In
particular, the following properties hold.

  1. (i)

For all $f\in\mathcal{B}^{n}$,

$f(\mathbf{z})=\sum_{\Omega\in\mathcal{P}_{n}}f_{\Omega,0}(\mathbf{z}_{\Omega}),\;\text{where}\quad
f_{\Omega,0}(\mathbf{z}_{\Omega})=\sum_{\Psi\subseteq\Omega}(-1)^{|\Omega|-|\Psi|}f_{\Psi}(\mathbf{z}_{\Psi})\in\mathcal{B}^{n}_{\Omega,0}.$

  2. (ii)

For each $\Omega\in\mathcal{P}_{n}$, the map $f\mapsto f_{\Omega,0}$ for
$\Omega\in\mathcal{P}_{n}$ is a contraction from $\mathcal{B}^{n}$ to
$\mathcal{B}^{|\Omega|}$.

  3. (iii)

For $f\in\mathcal{B}^{n}$, $\mathcal{D}_{n}f=0$ if and only if $f$ is a finite
sum of holomorphic functions with degree less than $n$. In that case,

$f(\mathbf{z})=\sum_{\Omega\in\mathcal{P}_{n},\Omega\neq
I_{n}}(-1)^{n-|\Omega|-1}f_{\Omega}(\mathbf{z}_{\Omega}).$

  4. (iv)

If $g\in H^{\infty}({\mathbb{C}}_{+}^{n})$, $\mathcal{D}_{n}g=0$, and
$g(\mathbf{z})=0$ whenever $z_{k}=\infty$ for at least one $k\in I_{n}$, then
$g=0$.

######  Example 2.8.

Let $\mu$ be a bounded Borel measure on ${\mathbb{R}}^{n}_{+}$, and
$\mathcal{L}\mu$ be its Laplace transform. The space $\mathcal{LM}^{n}$ of all
functions of the form $\mathcal{L}\mu$ is a Banach algebra in the norm
$\|\mathcal{L}\mu\|_{HP}=\|\mu\|$, and it is known as the Hille-Phillips
algebra. It is continuously included in $\mathcal{B}^{n}$ (see [6, Section
2.4] for the case $n=1$).

Let $f=\mathcal{L}\mu\in\mathcal{LM}^{n}$. Then the elementary decomposition
of $f$ is given by $f_{\Omega,0}=\mathcal{L}\mu_{\Omega}$, where
$\mu_{\Omega}$ is the restriction of $\mu$ to the subset

${\mathbb{R}}_{+,\Omega}:=\prod_{j=1}^{n}C_{j},\qquad
C_{j}:=\begin{cases}(0,\infty),\;&j\in\Omega,\\\
\\{0\\},&j\in\Omega^{c}.\end{cases}$

######  Remarks 2.9.

1\. In Propositions 2.3, 2.4, and 2.7, instead of letting
$\operatorname{Re}z_{j}$ tend to $\infty$, we could choose to take fixed
points $\zeta_{j}\in\overline{\mathbb{C}}_{+}$. The results remain valid. Such
a change would produce a different elementary decomposition of functions
$f\in\mathcal{B}^{n}$. However this is not a contradiction of the uniqueness,
because the spaces $\mathcal{B}^{n}_{\Omega,0}$ would also change.

2\. In the paper [24], Marti constructed a functional calculus for two
commuting closed operators under certain conditions. A subsequent short paper
by Caradus [11] showed that the conditions could be put in a better form and
pointed out that the construction could then be extended to $n$-tuples of
commuting operators in a natural way. Marti’s construction involved a
decomposition of a similar type to [21, Section 3.5], although the side
conditions are different. Caradus did not specify a decomposition but it would
undoubtedly have been of similar form to the elementary decomposition in
Proposition 2.7(i). Ichinose [20, p.235] also used a similar decomposition in
the context of two operators.

##  3\. Reproducing formulas and shifts

### Reproducing formulas

A reproducing formula for $f\in\mathcal{B}^{1}$ was given in [6, Proposition
2.20], and a version for $f\in\mathcal{B}^{2}$ was given in [21, Proposition
5.1.1]. In [8, Proposition 3.7], some formulas were obtained and the specific
formula with $s=1$ can be applied with $g=f^{\prime}$ if
$f\in\mathcal{B}^{1}$. We state that formula here.

######  Proposition 3.1.

Let $g$ be a holomorphic function on ${\mathbb{C}}_{+}$ such that

$\int_{{\mathbb{C}}_{+}}\frac{|g(z)|\,{\operatorname{Re}}\,z}{|z|^{2}}\,dS_{1}(z)<\infty.$

Define $Qg$ on ${\mathbb{C}}_{+}$ by

$(Qg)(z)=-\frac{2}{\pi}\int_{{\mathbb{C}}_{+}}g(\lambda)\frac{\operatorname{Re}\lambda}{(z+\overline{\lambda})^{2}}\,dS_{1}(\lambda).$

Then $Qg$ is holomorphic, and $g=(Qg)^{\prime}$.

We will present the corresponding statement for functions of $n$ variables in
Proposition 3.2. We use the following kernel functions:

$K(z,\lambda)=-\frac{2}{\pi(z+\lambda)^{2}},\;z,\lambda\in{\mathbb{C}}_{+},\qquad\mathcal{K}_{n}(\mathbf{z},{\bm{\lambda}})=\prod_{j=1}^{n}K(z_{j},\lambda_{j}),\;\mathbf{z},{\bm{\lambda}}\in{\mathbb{C}}_{+}^{n}.$

Let $g$ be a holomorphic function on ${\mathbb{C}}_{+}^{n}$ such that

(3.1)
$\int_{{\mathbb{C}}_{+}^{n}}|g({\bm{\lambda}})|\prod_{j=1}^{n}\frac{\operatorname{Re}\lambda_{j}}{|\lambda_{j}|^{2}}\,dS_{n}({\bm{\lambda}})<\infty.$

Define the function $\mathcal{Q}_{n}g$ on ${\mathbb{C}}_{+}^{n}$ by

$(\mathcal{Q}_{n}g)(\mathbf{z}):=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{K}_{n}(\mathbf{z},\overline{\bm{\lambda}})g({\bm{\lambda}})\prod_{j=1}^{n}\operatorname{Re}\lambda_{j}\,dS_{n}({\bm{\lambda}}),\qquad\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

This function is holomorphic on ${\mathbb{C}}_{+}^{n}$ and the derivatives can
be passed through the integral sign, the justification being that the
derivatives of the integrand with respect to the $\mathbf{z}$ variables can be
estimated locally uniformly by constant multiples of the integrand in (3.1).
Thus

$(\mathcal{D}_{n}Q_{n}g)(\mathbf{z})=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{D}_{n}{\mathcal{K}}_{n}(\mathbf{z},\overline{\bm{\lambda}})g({\bm{\lambda}})\,dV_{n}({\bm{\lambda}}),\qquad\mathbf{z}\in{\mathbb{C}}_{+}^{n},$

where the derivatives of ${\mathcal{K}}_{n}$ are with respect to the
$\mathbf{z}$ variables.

We now extend Proposition 3.1 to obtain the following reproducing formula for
functions on ${\mathbb{C}}_{+}^{n}$.

######  Proposition 3.2.

Let $g$ be a holomorphic function on ${\mathbb{C}}_{+}^{n}$ such that (3.1)
holds. Then $\mathcal{Q}_{n}g$ is holomorphic and

(3.2) $g=\mathcal{D}_{n}\mathcal{Q}_{n}g.$

###### Proof.

The proof is by induction, and the case $n=1$ is given in Proposition 3.1.

Let $n\geq 1$, and assume that (3.2) holds for $n$ variables. We consider now
the case of $n+1$ variables $(\mathbf{z},z_{n+1})$. Assume that
$g(\mathbf{z},z_{n+1})$ satisfies (3.1) for $n+1$ variables. For
$\mathbf{z}\in{\mathbb{C}}_{+}^{n}$, let
$h_{\mathbf{z}}(z_{n+1})=g(\mathbf{z},z_{n+1})$. By (3.1) and Fubini’s theorem
we obtain that

$\int_{{\mathbb{C}}_{+}}|h_{\mathbf{z}}(\lambda)|\frac{\operatorname{Re}{\lambda}}{|\lambda|^{2}}\,dS(\lambda)<\infty$

for almost all $\mathbf{z}\in{\mathbb{C}}_{+}^{n}$. For those $\mathbf{z}$, we
can apply Proposition 3.1 to the function $h_{\mathbf{z}}$. We obtain

$g(\mathbf{z},z_{n+1})=h_{\mathbf{z}}(z_{n+1})=\int_{{\mathbb{C}}_{+}}D_{n+1}K(z_{n+1},\lambda_{n+1})g(\mathbf{z},\lambda_{n+1})\,dV(\lambda_{n+1}).$

Now let $k_{z_{n+1}}(\mathbf{z})=g(\mathbf{z},z_{n+1})$. By applying Fubini’s
theorem and the inductive hypothesis to the functions $k_{z_{n+1}}$, we obtain
that, for almost all $z_{n+1}\in{\mathbb{C}}_{+}$,

$\displaystyle g(\mathbf{z},z_{n+1})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{D}_{n}{\mathcal{K}}_{n}(\mathbf{z},\overline{\bm{\lambda}})k_{z_{n+1}}({\bm{\lambda}})\,dV_{n}({\bm{\lambda}})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{D}_{n}{\mathcal{K}}_{n}(\mathbf{z},\overline{\bm{\lambda}})\int_{{\mathbb{C}}_{+}}D_{n+1}K(z_{n+1},\lambda_{n+1})g({\bm{\lambda}},\lambda_{n+1})\,dV(\lambda_{n+1})\,dV_{n}({\bm{\lambda}})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{n+1}}\mathcal{D}_{n+1}\mathcal{K}_{n+1}(\mathbf{z},z_{n+1},\overline{\bm{\lambda}}_{n},\overline{\lambda}_{n+1})g({\bm{\lambda}},\lambda_{n+1})\,dV_{n+1}({\bm{\lambda}},\lambda_{n+1})$
$\displaystyle=(\mathcal{D}_{n+1}Q_{n+1}g)(\mathbf{z},z_{n+1}).$

This establishes that $g=\mathcal{D}_{n+1}Q_{n+1}g$ almost everywhere. Since
both functions are holomorphic, it follows that they agree everywhere. ∎

######  Corollary 3.3.

Let $f\in\mathcal{B}^{n}$. Then
$\mathcal{D}_{n}f=\mathcal{D}_{n}\mathcal{Q}_{n}\mathcal{D}_{n}f$, and
$f_{\rm{el}}=\mathcal{Q}_{n}\mathcal{D}_{n}f$, so

(3.3)
$f_{\rm{el}}(\mathbf{z})=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{K}_{n}(\mathbf{z},\overline{\bm{\lambda}})(\mathcal{D}_{n}f)({\bm{\lambda}})\,dV_{n}({\bm{\lambda}}),\quad\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

In particular, if $f\in\mathcal{B}^{n}_{0}$, then
$f=\mathcal{Q}_{n}\mathcal{D}_{n}f$.

###### Proof.

Let $g=\mathcal{D}_{n}f$. Then

$\displaystyle\int_{{\mathbb{C}}_{+}^{n}}|g(\mathbf{z})|\prod_{j=1}^{n}\frac{\operatorname{Re}z_{j}}{|z_{j}|^{2}}\,dS_{n}(\mathbf{z})$
$\displaystyle\leq\int_{{\mathbb{R}}_{+}^{n}}(H_{I_{n}}f)({\bm{\alpha}})\int_{{\mathbb{R}}^{n}}\prod_{j=1}^{n}\frac{\alpha_{j}}{\alpha_{j}^{2}+t_{j}^{2}}\,d{\bm{\beta}}\,d{\bm{\alpha}}$
$\displaystyle=\pi^{n}\|f\|_{B^{n}_{I_{n}}}<\infty.$

Thus $g$ satisfies (3.1), so Proposition 3.2 shows that
$g=\mathcal{D}_{n}\mathcal{Q}_{n}g$.

By the dominated convergence theorem,
$(\mathcal{Q}_{n}\mathcal{D}_{n}f)(\mathbf{z})\to 0$ if
$\operatorname{Re}z_{j}\to\infty$ for any one $j$. Moreover,
$f_{\rm{el}}-\mathcal{Q}_{n}\mathcal{D}_{n}f$ is in
$H^{\infty}({\mathbb{C}}_{+}^{n})$ and in the kernel of $\mathcal{D}_{n}$ with
zero limits as $\operatorname{Re}z_{j}\to\infty$, and then it follows that
$f_{\rm{el}}=\mathcal{Q}_{n}\mathcal{D}_{n}f$ (see Proposition 2.7(iv)). ∎

######  Example 3.4.

In this example we establish the $\mathcal{B}_{k}$-norms and other properties
of various resolvent functions.

For $z=\alpha+i\beta\in{\mathbb{C}}_{+}$ and $\lambda\in{\mathbb{C}}_{+}$, let
$r_{z}(\lambda)=(\lambda+z)^{-1}$. Then $r_{z}\in\mathcal{B}^{1}_{0}$, and

$\|r_{z}^{k}\|_{\mathcal{B}_{0}^{1}}=\alpha^{-k},\quad\|r_{z}^{k}\|_{\mathcal{B}^{1}}=2\alpha^{-k},\quad
k\in{\mathbb{N}}.$

For $\mathbf{z}\in{\mathbb{C}}_{+}^{n}$ and
${\bm{\lambda}}\in{\mathbb{C}}_{+}^{n}$, let
$r_{\mathbf{z}}({\bm{\lambda}}):=\prod_{j=1}^{n}r_{z_{j}}(\lambda_{j}).$ Then
$\mathbf{r}_{\mathbf{z}}\in\mathcal{B}_{0}^{n}$, and

$\displaystyle\|\mathbf{r}_{\mathbf{z}}\|_{\mathcal{B}_{0}^{n}}$
$\displaystyle=\prod_{j=1}^{n}\|r_{z_{j}}\|_{\mathcal{B}_{0}^{1}}=\prod_{j=1}^{n}\alpha_{j}^{-1},\quad\|\mathbf{r}_{\mathbf{z}}^{k}\|_{\mathcal{B}_{0}^{n}}=\prod_{j=1}^{n}\|r_{z_{j}}^{k}\|_{\mathcal{B}_{0}^{1}}=\prod_{j=1}^{n}\alpha_{j}^{-k},$

where $\alpha_{j}=\operatorname{Re}z_{j}$ and $k\in{\mathbb{N}}$.

For $z,\zeta\in{\mathbb{C}}_{+}$,

$\|r_{z}-r_{\zeta}\|_{\mathcal{B}_{0}^{1}}=|z-\zeta|\,\|r_{z}r_{\zeta}\|_{\mathcal{B}_{0}^{1}}\leq|z-\zeta|\|r_{z}\|_{\mathcal{B}^{1}}\|\,\|r_{\zeta}\|_{\mathcal{B}^{1}}\leq\frac{4|z-\zeta|}{\operatorname{Re}z\operatorname{Re}\zeta}\to
0,$

as $z\to\zeta$. So the map $z\mapsto r_{z}$ is continuous from
${\mathbb{C}}_{+}$ to $\mathcal{B}_{0}^{1}$.

For $j\in I_{n}$, the map $\mathbf{z}\mapsto r_{z_{j}}$ is continuous from
${\mathbb{C}}_{+}^{n}$ to $\mathcal{B}_{0}^{1}$. Let
$r_{z,j}({\bm{\lambda}})=r_{z}(\lambda_{j})$. Then $\mathbf{z}\mapsto r_{z,j}$
is continuous from ${\mathbb{C}}_{+}^{n}$ to $\mathcal{B}^{n}$. Since
$\mathbf{r}_{\mathbf{z}}=\prod_{j=1}^{n}r_{z,j}$, the map
$\mathbf{z}\mapsto\mathbf{r}_{\mathbf{z}}$ is continuous from
${\mathbb{C}}_{+}^{n}$ to $\mathcal{B}_{0}^{n}$.

In fact, the map $\mathbf{z}\mapsto r(\mathbf{z})$ is holomorphic from
${\mathbb{C}}_{+}^{n}$ to $\mathcal{B}_{0}^{n}$, but we do not need this.

### Shifts

We will use shifts of the variables $z_{j}$ for the construction of the
$\mathcal{B}$-calculus for operators. In [6] and [21], we used shifts not only
in the horizontal direction but also in the vertical directions. In this
paper, we define the functional calculus by a different route, and we do not
need to use the vertical shifts. So we present two lemmas here about the
horizontal shifts on $\mathcal{B}^{n}$.

First we will establish a variant of the reproducing formula (3.3) for shifted
functions in $\mathcal{B}_{0}^{n}$. This result is a special case of [6, Lemma
2.18(2)] for $n=1$, and [21, Proposition 5.4.1] for $n=2$.

######  Lemma 3.5.

If $f\in\mathcal{B}^{n}_{0},$ then for every
${\bm{t}}\in{\mathbb{R}}_{+}^{n}$,

(3.4)
$f(\mathbf{z}+{\bm{t}})=\int_{{\mathbb{C}}^{n}_{+}}\mathcal{K}_{n}(\mathbf{z}+{{\bm{t}}},\overline{{\bm{\lambda}}})(\mathcal{D}_{n}f)({\bm{\lambda}})\,dV_{n}({\bm{\lambda}}),\qquad\mathbf{z}\in{\mathbb{C}}^{n}_{+}.$

###### Proof.

We begin with the case when $n=1$. From Corollary 3.3,

$f(z+t)=-\frac{2}{\pi}\int_{0}^{\infty}\alpha\int_{{\mathbb{R}}}\frac{f^{\prime}(\alpha+t+i\beta)}{(z+\alpha-i\beta)^{2}}\,d\beta\,d\alpha,\qquad
z\in{\mathbb{C}}_{+}.$

Changing variables in the inner integral, noting that
$f^{\prime}(\alpha+\lambda)$ is bounded for $\lambda\in{\mathbb{C}}_{+}$ and a
fixed $\alpha>0$, and using Cauchy’s theorem, we obtain

$\displaystyle\int_{{\mathbb{R}}}\frac{f^{\prime}(\alpha+t+i\beta)}{(z+\alpha-i\beta)^{2}}\,d\beta$
$\displaystyle=-i\int_{t+i{\mathbb{R}}}\frac{f^{\prime}(\alpha+\lambda)}{(z+\alpha+t-\lambda)^{2}}\,d\lambda$
$\displaystyle=-i\int_{i{\mathbb{R}}}\frac{f^{\prime}(\alpha+\lambda)}{(z+\alpha+t-\lambda)^{2}}\,d\lambda=\int_{{\mathbb{R}}}\frac{f^{\prime}(\alpha+i\beta)}{(z+\alpha+t-i\beta)^{2}}\,d\beta.$

So, we obtain the formula (3.5) for $n=1$:

$f(z+t)=-\frac{2}{\pi}\int_{0}^{\infty}\alpha\int_{{\mathbb{R}}}\frac{f^{\prime}(\alpha+i\beta)}{(z+t+\alpha-i\beta)^{2}}\,d\beta\,d\alpha,\qquad
z\in{\mathbb{C}}_{+},\,t>0.$

For $n>1$, we may consider the individual variables one at a time, applying
the same argument as above to show that

$\hskip
40.0pt\int_{{\mathbb{C}}^{n}_{+}}\mathcal{K}_{n}(\mathbf{z}+{{\bm{t}}^{j-1}},\overline{{\bm{\lambda}}})(\mathcal{D}_{n}f)({\bm{\lambda}}+{\bm{t}}_{j})\,dV_{n}({\bm{\lambda}})\\\
=\int_{{\mathbb{C}}^{n}_{+}}\mathcal{K}_{n}(\mathbf{z}+{{\bm{t}}^{j}},\overline{{\bm{\lambda}}})(\mathcal{D}_{n}f)({\bm{\lambda}}+{\bm{t}}_{j+1})\,dV_{n}({\bm{\lambda}}),\hskip
40.0pt$

where

${\bm{t}}^{j}=(t_{1},\dots,t_{j},0,0,\dots,0),\;{\bm{t}}_{j}=(0,\dots,0,t_{j},t_{j+1},\dots,t_{n}),\quad
j\in I_{n}.\qed$

Next we show that horizontal shifts are strongly continuous on
$\mathcal{B}^{n}$. For $f\in\mathcal{B}^{n}$,
${\bm{t}}=(t_{1},\dots,t_{n})\in{\mathbb{R}}_{+}^{n}$, and
$\mathbf{z}\in\mathbb{C}_{+}^{n}$, let

(3.5) $(T({\bm{t}})f)({\mathbf{z}}):=f(\mathbf{z}+{\bm{t}}).$

It is easy to see that $T({\bm{t}})$ is a contraction on
$(\mathcal{B}^{n},\|\cdot\|_{\mathcal{B}^{n}})$. The following lemma was
proved in [6, Lemma 2.6] for $n=1$ and in [21, Proposition 4.2.2] for $n=2$.

######  Lemma 3.6.

The map $T:{\mathbb{R}}_{+}^{n}\to L(\mathcal{B}^{n})$ is strongly continuous.

###### Proof.

Let

$(T_{j}(t)f)({\mathbf{z}}):=f(z_{1},\dots,z_{j}+t,\dots,z_{n}),\qquad j\in
I_{n},\,t\geq 0.$

Then

$T({\bm{t}})=T_{1}(t_{1})\dots T_{n}(t_{n}).$

It suffices to show that the semigroups $(T_{j}(t))_{t\geq 0}$ are continuous
in the strong operator topology on $L(\mathcal{B}^{n})$. Without loss of
generality, we will take $j=1$, and we will write
$\mathbf{z}=(z_{1},\mathbf{z}^{\prime})\in{\mathbb{C}}_{+}^{n}$.

Let $f\in\mathcal{B}^{n}$ and $\Omega\in\mathcal{P}_{n}$ with $k=|\Omega|$. We
need to show that

(3.6) $\lim_{t\to 0}\|T_{1}(t)f-f\|_{\mathcal{B}^{n}_{\Omega}}=0.$

There are two cases: (1) $1\in\Omega$, (2) $1\notin\Omega$.

Case 1: $1\in\Omega$. Let
${\bm{\alpha}}_{\Omega}=({\bm{\alpha}}_{j})_{j\in\Omega}\in(0,\infty)^{k}$,
and let $\mathbf{z}\in W_{{\bm{\alpha}}_{\Omega}}$. By an extended version of
Cauchy’s inequality,

(3.7)
$\left|(D_{1}\mathcal{D}_{\Omega}f)(\mathbf{z})\right|\leq\frac{2^{-(k-1)}\|f\|_{\infty}}{\alpha_{1}\prod_{j\in\Omega}\alpha_{j}}.$

This estimate is valid for $\operatorname{Re}z_{1}>\alpha_{1}$, by the maximum
principle. Hence

$\displaystyle\left|(\mathcal{D}_{\Omega}f)(z_{1}+t,\mathbf{z}^{\prime})-(\mathcal{D}_{\Omega}f)(z_{1},\mathbf{z}^{\prime})\right|\leq\int_{0}^{t}\left|(D_{1}\mathcal{D}_{\Omega}f)(z_{1}+s,\mathbf{z}^{\prime})\right|\,ds$
$\displaystyle\leq\frac{2^{k-1}t\|f\|_{\infty}}{\alpha_{1}\prod_{j\in\Omega}\alpha_{j}}.$

Taking the supremum over all possible choices of $\mathbf{z}\in
W_{{\bm{\alpha}}_{\Omega}}$, this establishes that

$H_{\Omega}(T_{1}(t)f-f)({\bm{\alpha}}_{\Omega})\leq\frac{2^{-(k-1)}t\|f\|_{\infty}}{\alpha_{1}\prod_{j\in\Omega}\alpha_{j}}\to
0,\quad t\to 0.$

In addition,

(3.8)
$H_{\Omega}(T_{1}(t)f-f)({\bm{\alpha}}_{\Omega})\leq(H_{\Omega}T_{1}(t)f)({\bm{\alpha}}_{\Omega})+H_{\Omega}f({\bm{\alpha}}_{\Omega})\leq
2H_{\Omega}f({\bm{\alpha}}_{\Omega}),$

by the maximum principle. Hence, the dominated convergence theorem implies
that (3.6) holds.

Case 2: $1\notin\Omega$. Let $\Omega^{\prime}=\Omega\cup\\{1\\}$,
${\bm{\alpha}}_{\Omega}=({\bm{\alpha}}_{j})_{j\in\Omega}\in(0,\infty)^{k}$,
and $\mathbf{z}\in W_{{\bm{\alpha}}_{\Omega}}$. Then
$\mathbf{z}_{\Omega^{\prime}}=(z_{1},\mathbf{z}_{\Omega})$. The definition of
$H_{\Omega^{\prime}}$ and its monotonicity show that

$\left|(\mathcal{D}_{\Omega}f)(z_{1}+t,\mathbf{z}^{\prime})-(D_{\Omega}f)(z_{1},\mathbf{z}^{\prime})\right|\leq\int_{0}^{t}\left|(\mathcal{D}_{\Omega^{\prime}}f)(z_{1}+s,\mathbf{z}^{\prime})\right|\,ds\\\
\leq\int_{0}^{t}(H_{\Omega^{\prime}}f)(\alpha_{1}+s,{\bm{\alpha}}_{\Omega})\,ds\leq\int_{0}^{t}(H_{\Omega^{\prime}}f)(s,{\bm{\alpha}}_{\Omega})\,ds.$

The final expression depends only on $f$, $t$ and ${\bm{\alpha}}_{\Omega}$, so
we may take the supremum over all $\mathbf{z}\in W_{{\bm{\alpha}}_{\Omega}}$
and infer that

(3.9)
$H_{\Omega}(T_{1}(t)f-f)({\bm{\alpha}}_{\Omega})\leq\int_{0}^{t}(H_{\Omega^{\prime}}f)(s,{\bm{\alpha}}_{\Omega})\,ds.$

Since $\|f\|_{\mathcal{B}^{n}_{\Omega^{\prime}}}$ is finite,
$H_{\Omega^{\prime}}f$ is integrable over ${\mathbb{R}}_{+}^{k+1}$. Fubini’s
theorem implies that

$\alpha_{1}\mapsto\int_{{\mathbb{R}}_{+}^{k}}(H_{\Omega^{\prime}}f)(\alpha_{1},{\bm{\alpha}}_{\Omega})\,d{\bm{\alpha}}_{\Omega}$

is integrable over ${\mathbb{R}}_{+}$, and then (3.9) and Fubini’s theorem
imply that

$\displaystyle\|T_{1}(t)f-f\|_{\mathcal{B}^{n}_{\Omega}}$
$\displaystyle=\int_{{\mathbb{R}}_{+}^{k}}H_{\Omega}(T_{1}(t)f-f)({\bm{\alpha}}_{\Omega})\,d{\bm{\alpha}}_{\Omega}$
$\displaystyle\leq\int_{0}^{t}\int_{{\mathbb{R}}_{+}^{k}}(H_{\Omega^{\prime}}f)(\alpha_{1},{\bm{\alpha}}_{\Omega})\,d{\bm{\alpha}}_{\Omega}\,d\alpha_{1}\to
0,\qquad t\to 0.\qed$

######  Remarks 3.7.

1\. Case 2 of the proof above can be applied when $\Omega$ is empty, to show
that the left shifts are strongly continuous in the $H^{\infty}$-norm, and
hence that $f$ is uniformly continuous, as shown in Proposition 2.1.

2\. For $n=2$, the proof above differs from [21, Proposition 4.2.2], where the
proof relies on switching to an equivalent norm on $\mathcal{B}^{2}$. That
approach can also be applied for $n>2$. It simplifies the proof of case 2
above, provided one accepts the equivalence of norms.

3\. It was shown in [6, Proposition 4.6.2] that the shift semigroup on
$\mathcal{B}^{1}$ extends to a bounded holomorphic semigroup of angle $\pi/2$.

### Sums of variables

In this section, we consider how very simple changes of the variables $z_{j}$
preserve $\mathcal{B}^{n}$. As noted in [6, Lemma 2.6(5)] a trivial scaling of
variables shows that if $f\in\mathcal{B}^{1}$, $b>0$ and $g(z)=f(bz)$, then
$g\in\mathcal{B}^{1}$, and $\|g\|_{\mathcal{B}^{1}}=\|f\|_{\mathcal{B}^{1}}$,
and the same applies for functions in $\mathcal{B}^{n}$ with scaling of some
or all variables.

In addition, the family of algebras $\mathcal{B}^{n}$ is preserved by summing
some or all of the variables $z_{j}$. In order to avoid heavy formulas, we
present this only in the case where all the variables are summed. For $n=2$,
the result was proved in [21, Lemma 3.8.1].

######  Proposition 3.8.

Let $f\in\mathcal{B}^{1}$, $n\in{\mathbb{N}}$, and define $g$ on
${\mathbb{C}}_{+}^{n}$ by

(3.10)
$g(\mathbf{z})=f(z_{1}+\dots+z_{n}),\qquad\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

Then $g\in\mathcal{B}^{n}$, and the map $f\mapsto g$ from $\mathcal{B}^{1}$ to
$\mathcal{B}^{n}$ is bounded.

###### Proof.

We may assume that $f\in\mathcal{B}^{1}_{0}$. Clearly $g$ is holomorphic on
${\mathbb{C}}_{+}^{n}$ and $g(\mathbf{z})\to 0$ as
$\operatorname{Re}z_{j}\to\infty$. By Proposition 2.6, it suffices to show
that $\|g\|_{\mathcal{B}^{n}_{0}}<\infty$. We have

$\|g\|_{\mathcal{B}^{n}_{0}}=\int_{{\mathbb{R}}^{n}_{+}}\sup_{\beta\in{\mathbb{R}}}\left|f^{(n)}(\alpha_{1}+\dots+\alpha_{n}+i\beta)\right|\,d{\bm{\alpha}}.$

The change of variables

$t_{k}=\sum_{j=1}^{k}\alpha_{j},\qquad k\in I_{n},$

establishes that

$\displaystyle\|g\|_{\mathcal{B}^{n}_{0}}$
$\displaystyle=\int_{S_{n}}\sup_{\beta\in{\mathbb{R}}}\left|f^{(n)}(t_{n}+i\beta)\right|\,d{\bm{t}},$

where $S_{n}=\\{(t_{k})_{k\in I_{n}}\in{\mathbb{R}}_{+}^{n}:0\leq t_{1}\leq
t_{2}\leq\dots\leq t_{n}\\}$. Integrating with respect to
$t_{1},\dots,t_{n-1}$ and replacing $t_{n}$ by $t$ shows that

$\|g\|_{\mathcal{B}^{n}_{0}}=\int_{0}^{\infty}\frac{t^{n-1}}{(n-1)!}\sup_{\beta\in{\mathbb{R}}}\left|f^{(n)}(t+i\beta)\right|\,dt.$

A higher-order version of Cauchy’s inequality gives

$t^{n-1}\sup_{y\in{\mathbb{R}}}|f^{(n)}(t+i\beta)|\leq
2^{n-1}C_{n-1}\sup_{\beta\in{\mathbb{R}}}|f^{\prime}({t/2+i\beta})|,$

where $C_{n-1}$ is as in [6, Lemma 2.1(4)]. Hence

$\|g\|_{\mathcal{B}_{0}^{n}}\leq\frac{2^{n-1}C_{n-1}}{(n-1)!}\int_{0}^{\infty}\sup_{\beta\in{\mathbb{R}}}|f^{\prime}(t/2+i\beta)|\,dt=\frac{2^{n}C_{n-1}}{(n-1)!}\|f\|_{\mathcal{B}_{0}^{1}}.$

This establishes that $g\in\mathcal{B}_{0}^{n}$, and also that the map
$f\mapsto g$ is bounded from $\mathcal{B}^{1}$ to $\mathcal{B}^{n}$. ∎

It is clear that if $g$ as defined in (3.10) is in $\mathcal{B}^{n}$, then
$f\in\mathcal{B}^{1}$, since $f(z)=g(z,0,0,\dots,0)$.

##  4\. The $\mathcal{B}^{n}$-calculus

Let ${\mathcal{A}}:=(A_{1},\dots,A_{n})$ be an $n$-tuple of operators on a
Banach space $X$. We assume that the spectrum of each operator $A_{j}$ is
contained in $\overline{\mathbb{C}}_{+}$ and their resolvents commute with
each other. Let $\Omega\in\mathcal{P}_{n}$. We will say that ${\mathcal{A}}$
satisfies the (GSFΩ) condition if the following holds for all $x\in X$, and
$x^{*}\in X^{*}$:

$\sup_{\alpha_{j}>0,j\in\Omega}\int_{{\mathbb{R}}^{|\Omega|}}\big{|}\langle\prod_{j\in\Omega}\alpha_{j}(A_{j}+\alpha_{j}-i\beta_{j})^{-2}x,x^{*}\rangle\big{|}\,d{\bm{\beta}}<\infty.$

By the Closed Graph theorem, if (GSFΩ) holds, there is a constant
$\gamma_{{\mathcal{A}}_{\Omega}}$ such that

(4.1)
$\int_{{\mathbb{R}}^{|\Omega|}}\big{|}\langle\prod_{j\in\Omega}\alpha_{j}(A_{j}+\alpha_{j}-i\beta_{j})^{-2}x,x^{*}\rangle\big{|}\,d{\bm{\beta}}\leq\gamma_{{\mathcal{A}}_{\Omega}}\|x\|\,\|x^{*}\|$

for all $x\in X$ and $x^{*}\in X^{*}$. If $\Omega$ is empty, then the
condition is deemed to be satisfied. For $\Omega=\\{j\\}$, the condition says
that $A_{j}$ satisfies the (GSF) condition for the single operator $A_{j}$ as
in [6, Section 4.1]. If $\Omega=I_{n}$, we will say that ${\mathcal{A}}$
satisfies the (GSFn) condition and we will write $\gamma_{\mathcal{A}}$
instead of $\gamma_{I_{n}}$.

We will say that ${\mathcal{A}}$ satisfies the full (GSF) condition if
${\mathcal{A}}$ satisfies (GSFΩ) for all $\Omega\in\mathcal{P}_{n}$. For
$n=2$, the full (GSF) condition agrees with [21, Section 6.1].

######  Definition 4.1.

A $\mathcal{B}^{n}$-calculus for ${\mathcal{A}}$ is a bounded algebra
homomorphism $\Phi:\mathcal{B}^{n}\to L(X)$ such that
$\Phi(r_{\lambda,j})=(A_{j}+\lambda)^{-1}$ for all $j\in I_{n}$ and
$\lambda\in{\mathbb{C}}_{+}$.

There is a similar definition of a $\mathcal{B}_{0}^{n}$-caclulus. In this and
the next section, we will show that the (GSF) condition is a sufficient
condition for ${\mathcal{A}}$ to have a $\mathcal{B}^{n}$-calculus, but we
first show that the (GSF) condition is necessary. This was proved in [7,
Theorem 6.1] for $n=1$ and in [21, Theorem 6.3.1] for $n=2$. The proof below
is a variant of those proofs.

We will write
$\mathbf{r}_{\mathbf{z}}({\mathcal{A}}):=\prod_{j=1}^{n}(A_{j}+z_{j})^{-1}.$

######  Proposition 4.2.

Let ${\mathcal{A}}$ be an $n$-tuple of operators on a Banach space $X$, and
assume that the spectrum of each operator $A_{j}$ is contained in
$\overline{\mathbb{C}}^{n}_{+}$, their resolvents commute with each other, and
there is a $\mathcal{B}_{0}^{n}$-calculus $\Phi$ for ${\mathcal{A}}$. Then
${\mathcal{A}}$ satisfies the (GSFn) condition.

###### Proof.

Let $\varphi:{\mathbb{R}}^{n}\to{\mathbb{C}}$ be a continuous function with
compact support, and ${\bm{\alpha}}\in(0,\infty)^{n}$. As shown in Example
3.4, the map ${\bm{\beta}}\mapsto\mathbf{r}^{2}_{{\bm{\alpha}}-i{\bm{\beta}}}$
is continuous from $(0,\infty)^{n}$ to $\mathcal{B}_{0}^{n}$ and
$\|\mathbf{r}^{2}_{{\bm{\alpha}}-i{\bm{\beta}}}\|_{\mathcal{B}_{0}^{n}}\leq
4^{n}\prod_{j=1}^{n}\alpha_{j}^{-1}$. Hence

$G_{{\bm{\alpha}},\varphi}:=\int_{{\mathbb{R}}^{n}}\varphi({\bm{\beta}})\mathbf{r}_{{\bm{\alpha}}-i{\bm{\beta}}}^{2}\,d{\bm{\beta}},$

exists as a Bochner integral with values in $\mathcal{B}_{0}^{n}$. Since
$\Phi$ is a bounded linear operator,

$\Phi(G_{{\bm{\alpha}},\varphi})=\int_{{\mathbb{R}}^{n}}\varphi({\bm{\beta}})\mathbf{r}_{{\bm{\alpha}}-i{\bm{\beta}}}({\mathcal{A}})^{2}\,d{\bm{\beta}}$

and hence, for unit vectors $x\in X$ and $x^{*}\in X^{*}$,

$\prod_{j=1}^{n}\alpha_{j}\left|\int_{{\mathbb{R}}^{n}}\varphi({\bm{\beta}})\langle\mathbf{r}_{{\bm{\alpha}}-i{\bm{\beta}}}({\mathcal{A}})^{2}x,x^{*}\rangle\,d{\bm{\beta}}\right|\leq
4^{n}\|\varphi\|_{\infty}\|\Phi\|.$

Since this holds for all continuous functions $\varphi$ with compact support,
it follows that (GSFn) holds. ∎

######  Corollary 4.3.

Let ${\mathcal{A}}$ be an $n$-tuple of operators on a Banach space $X$, and
assume that the spectrum of each operator $A_{j}$ is contained in
$\overline{\mathbb{C}}_{+}^{n}$, their resolvents commute with each other, and
there is a bounded algebra homomorphism $\Phi:\mathcal{B}^{n}\to L(X)$ such
that $\Phi(\mathbf{r}_{\mathbf{z}})=\mathbf{r}_{\mathbf{z}}({\mathcal{A}})$
for all $\mathbf{z}\in{\mathbb{C}}_{+}^{n}$. Then ${\mathcal{A}}$ satisfies
the full (GSF) condition.

###### Proof.

For $\Omega\in\mathcal{P}_{n}$, we may deduce that (GSFΩ) holds, by applying
Proposition 4.2 to $\mathcal{B}^{\Omega}_{0}$ and the restriction of $\Phi$ to
that algebra. ∎

Now we give two classes of operators where ${\mathcal{A}}$ satisfies the full
(GSF) condition for all values of $n$. Proofs were given in [6] for $n=1$ and
in [21] for $n=2$, and the general proofs are minor variants of those cases.

######  Examples 4.4.

1\. Let $-A_{j}$ be the generator of a bounded $C_{0}$-semigroup
$(e^{-tA_{j}})_{t\geq 0}$ on a Hilbert space $X$, for each $j\in I_{n}$, and
let $K_{j}=\sup_{t\geq 0}\|e^{-tA_{j}}\|$. Assume that the semigroups commute
with each other (or equivalently, the resolvents of $A_{j}$ commute with each
other). Then ${\mathcal{A}}$ satisfies the full (GSF) condition, with
$\gamma_{\mathcal{A}}=2^{n}\prod_{j=1}^{n}K_{j}^{2}$. This follows from
applications of Plancherel’s theorem and the Cauchy-Schwarz inequality. See
the proofs for $n=1$ in [6, Examples 4.1] and for $n=2$ in [21, Example
6.1.1].

2\. Let $A_{j}\,(j\in I_{n})$ be sectorial operators of angle less than
$\pi/2$ on a Banach space $X$, and assume that the resolvents commute with
each other (equivalently, $-A_{j}$ is the generator of a bounded holomorphic
$C_{0}$-semigroup $(e^{-tA_{j}})_{t\geq 0}$ on a Banach space $X$, for $j\in
I_{n}$, and the semigroups commute with each other). Let
$M_{j}=\sup_{z\in{\mathbb{C}}_{+}}\|z(A_{j}+z)^{-1}\|$. Then ${\mathcal{A}}$
satisfies the full (GSF) condition, with
$\gamma_{\mathcal{A}}=2^{n}\prod_{j=1}^{n}M_{j}^{2}$. This follows from
standard resolvent estimates for sectorial operators. See the proofs for $n=1$
in [6, Section 4.2] and for $n=2$ in [21, Example 6.1.2].

Now, assume that ${\mathcal{A}}$ satisfies the (GSFn) condition, and let
$f\in\mathcal{B}_{0}^{n}$. Recall from Corollary 3.3 that the following
reproducing formula holds:

(4.2)
$f(\mathbf{z})=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{K}_{n}(\mathbf{z},\overline{{\bm{\lambda}}})(\mathcal{D}_{n}f)({\bm{\lambda}})\,dV_{n}({\bm{\lambda}}),\qquad\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

We define $f({\mathcal{A}})$ by the corresponding formula:

(4.3) $\langle
f({\mathcal{A}})x,x^{*}\rangle:=\int_{{\mathbb{C}}_{+}^{n}}\langle\mathcal{K}_{n}({\mathcal{A}},\overline{{\bm{\lambda}}})x,x^{*}\rangle\,(\mathcal{D}_{n}f)({\bm{\lambda}})\,dV_{n}({\bm{\lambda}}),$

where

(4.4)
${\mathcal{K}}_{n}({\mathcal{A}},\overline{{\bm{\lambda}}}):=\left(-\frac{2}{\pi}\right)^{n}\mathbf{r}_{\overline{\bm{\lambda}}}({\mathcal{A}})^{2}=\left(-\frac{2}{\pi}\right)^{n}\prod_{j=1}^{n}(A_{j}+\overline{\lambda}_{j})^{-2},\qquad{\bm{\lambda}}\in{\mathbb{C}}_{+}^{n}.$

By the (GSFn) condition, the integral is absolutely convergent, and the
formula (4.3) defines $f({\mathcal{A}})$ as a map in $L(X,X^{**})$, with norm
at most $\gamma_{\mathcal{A}}\|f\|_{\mathcal{B}^{n}_{0}}$. In order to obtain
a $\mathcal{B}^{n}$-calculus for ${\mathcal{A}}$, we need to extend the
definition of $f({\mathcal{A}})$ to functions $f\in\mathcal{B}^{n}$, show that
$f({\mathcal{A}})$ maps $X$ into $X$ for all $f\in\mathcal{B}^{n}$, and show
that the map $f\mapsto f({\mathcal{A}})$ is an algebra homomorphism from
$\mathcal{B}^{n}$ to $L(X)$.

If the operators $A_{j}$ are the negative generators of $n$ commuting
$C_{0}$-semigroups, and $f=\mathcal{L}\mu$ for some bounded measure $\mu$ on
${\mathbb{R}}_{+}^{n}$, then $f({\mathcal{A}})$ as defined above agrees with
the Hille-Phillips calculus for $\mathcal{LM}^{n}$. This was shown in [6,
Lemma 4.2] for $n=1$ and in [21, Lemma 6.1.4] for $n=2$, . The proof for
$n\geq 3$ is similar, using Example 2.8 and higher-order integrals. A
consequence of this is that the definition of
${\mathcal{K}}_{n}({\mathcal{A}},\overline{\bm{\lambda}})$ in (4.4) agrees
with the definition obtained by putting
$f(\mathbf{z})={\mathcal{K}}_{n}(\mathbf{z},\overline{\bm{\lambda}})$ in
(4.3), for a fixed ${\bm{\lambda}}$.

Now we assume that ${\mathcal{A}}$ satisfies the full (GSF) condition. If $f$
is an elementary function in $\mathcal{B}^{n}$, with order $k$ and support
$\Omega$, we can define $f({\mathcal{A}})$ by applying the
$\mathcal{B}_{0}^{k}$-calculus for
${\mathcal{A}}_{\Omega}:=(A_{j})_{j\in\Omega}$ to the function
$f_{\Omega}\in\mathcal{B}_{0}^{k}$, so

(4.5) $\displaystyle\langle f(\mathcal{A})x,x^{*}\rangle$
$\displaystyle:=\int_{{\mathbb{C}}_{+}^{k}}\langle\mathcal{K}_{k}(\mathcal{A}_{\Omega},\overline{{\bm{\lambda}}})x,x^{*}\rangle\,(\mathcal{D}_{k}f_{\Omega})({\bm{\lambda}})\,dV_{k}({\bm{\lambda}})$

for all $x\in X$ and $x^{*}\in X^{*}$. In particular, if $\Omega=\\{j\\}$ for
some $j\in I_{n}$, $g\in\mathcal{B}^{1}$, and we define
$f(\mathbf{z})=g(z_{j})$, then we recover the $\mathcal{B}^{1}$-calculus for
the operator $A_{j}$. Thus this definition establishes that
$f({\mathcal{A}})=(A_{j}+\lambda)^{-1}$ if $f(\mathbf{z})=r_{\lambda}(z_{j})$.

Now let $f$ be an arbitrary function in $\mathcal{B}^{n}$. To define
$f({\mathcal{A}})$ we use the elementary decomposition of $f$ as in
Proposition 2.7(i). We define

(4.6)
$f({\mathcal{A}})=\sum_{\Omega\in\mathcal{P}_{n}}f_{\Omega,0}({\mathcal{A}}_{\Omega}),$

Then we define a bounded linear map from $\Phi_{A}:\mathcal{B}^{n}\to
L(X,X^{**})$ by

(4.7) $\Phi_{A}(f)=f({\mathcal{A}}).$

Then
$\|\Phi_{\mathcal{A}}\|\leq\gamma_{\mathcal{A}}:=\max_{\Omega\in\mathcal{P}_{n}}\gamma_{{\mathcal{A}}_{\Omega}}$,
where $\gamma_{{\mathcal{A}}_{\Omega}}$ is as in (4.1). The norm of the
restriction of $\Phi_{A}$ to $\mathcal{B}_{0}^{n}$ is the minimal value of
$\gamma_{{\mathcal{A}}_{n}}$, and we will denote it by
$\gamma_{{\mathcal{A}},0}$.

In the next section, we will prove that $\Phi_{A}$ is an algebra homomorphism
from $\mathcal{B}^{n}$ to $L(X)$. It will suffice to show that
$f({\mathcal{A}})\in L(X)$ and
$(fg)({\mathcal{A}})=f({\mathcal{A}})g({\mathcal{A}})$ for all elementary
functions $f$ and $g$ in $\mathcal{B}^{n}$. These properties were proved in
[6] for $n=1$ and in [21] for $n=2$, by a rather complicated method, and the
method used in Section 5 is simpler, and illuminating even in the case $n=1$.

Before moving on to the main proof, we present two lemmas. The first one may
help when considering (4.6).

######  Lemma 4.5.

Assume that ${\mathcal{A}}=(A_{1},\dots,A_{n})$ satisfies the full (GSF)
condition, and let ${\mathcal{A}}_{n-1}=(A_{1},\dots,A_{n-1})$.

  1. 1.

For $f\in\mathcal{B}^{n}$ and $\Omega\in\mathcal{P}_{n}$ with $|\Omega|=k$,

(4.8) $\langle
f_{\Omega,0}({\mathcal{A}})x,x^{*}\rangle:=\int_{{\mathbb{C}}_{+}^{k}}\langle\mathcal{K}_{k}({\mathcal{A}}_{\Omega},\overline{{\bm{\lambda}}})x,x^{*}\rangle\,(\mathcal{D}_{\Omega}f_{\Omega})({\bm{\lambda}})\,dV_{k}({\bm{\lambda}}).$

  2. 2.

Let $f\in\mathcal{B}^{n}_{0}$. Then
$(D_{n}f)(\cdot,\lambda_{n})\in\mathcal{B}^{n-1}_{0}$ for all
$\lambda_{n}\in{\mathbb{C}}_{+}$, and, for all $x\in X$ and $x^{*}\in X^{*}$,

$\langle f({\mathcal{A}})x,x^{*}\rangle=\int_{{\mathbb{C}}_{+}}\langle
K(A_{n},\overline{\lambda}_{n})(D_{n}f)({\mathcal{A}}_{n-1},\lambda_{n})x,x^{*}\rangle\,dV(\lambda_{n}),$

###### Proof.

The first statement is proved in Corollary 3.3.

Proposition 2.3, with $\Omega=I_{n-1}$ and $\Psi=\\{n\\}$, establishes that
$(D_{n}f)(\cdot,\lambda_{n})\in\mathcal{B}^{n-1}_{0}$ for all
$\lambda_{n}\in{\mathbb{C}}_{+}$. Fubini’s theorem shows that

$\displaystyle\langle f(\mathcal{A})x,x^{*}\rangle$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{n}}\langle{\mathcal{K}}_{n-1}({\mathcal{A}}_{n-1},\overline{{\bm{\lambda}}}_{n-1})K(A_{n},\overline{\lambda}_{n})x,x^{*}\rangle\,(\mathcal{D}_{n}f)({\bm{\lambda}}_{n-1},\lambda_{n})\,dV_{n}({\bm{\lambda}}_{n-1},\lambda_{n})$
$\displaystyle=\int_{{\mathbb{C}}_{+}}\langle
K(A_{n},\overline{\lambda}_{n})(D_{n}f)({\mathcal{A}}_{n-1},\lambda_{n})x,x^{*}\rangle\,dV(\lambda_{n}),$

for all $x\in X$ and $x^{*}\in X^{*}$. ∎

In order to justify some applications of Fubini’s theorem, and differentiation
through integral signs, in the next section, we will apply estimates for
$n$-tuples of operators of the form
${\mathcal{A}}+{\bm{t}}:=(A_{j}+t_{j})_{j\in I_{n}}$, where $t_{j}>0$, and we
will use the following fact. For $n=1$, it is a special case of [6, Corollary
4.5]. Here $(T({\bm{t}}))_{{\bm{t}}\in{\mathbb{R}}_{+}^{n}}$ are the shifts as
defined in (3.5).

######  Lemma 4.6.

Assume that ${\mathcal{A}}$ satisfies the (GSFn) condition. Let
$f\in\mathcal{B}_{0}^{n}$ and ${\bm{t}}\in{\mathbb{R}}_{+}^{n}$. Then

$f(\mathcal{A}+{\bm{t}})=(T({\bm{t}})f)(\mathcal{A}).$

This can easily be proved by the same method as in Lemma 3.5, with
${\mathcal{K}}_{n}(\mathbf{z},\overline{\bm{\lambda}})$ being replaced by
$\langle{\mathcal{K}}_{n}({\mathcal{A}},\overline{\bm{\lambda}})x,x^{*}\rangle$,
${\mathcal{K}}_{n}(\mathbf{z}+{\bm{t}},\overline{\bm{\lambda}})$ by
$\langle{\mathcal{K}}_{n}({\mathcal{A}}+{\bm{t}},\overline{\bm{\lambda}})x,x^{*}\rangle$,
and $(\mathcal{D}_{n}f)({\bm{\lambda}})$ by
$(\mathcal{D}_{n}f)({\bm{\lambda}}+{\bm{t}})$.

##  5\. The main properties for $\mathcal{B}^{n}$

In this section, we assume that $\mathcal{A}:=(A_{1},\dots,A_{n})$ is an
$n$-tuple of operators which satisfy the full (GSF) condition on a Banach
space $X$, and we denote by $\Phi_{\mathcal{A}}$ the map $f\mapsto
f({\mathcal{A}})$ from $\mathcal{B}_{0}^{n}$ to $L(X,X^{**})$, as defined in
(4.5), (4.6) and (4.7).

We will establish three properties:

  * •

the operator $f({\mathcal{A}})$ maps $X$ into $X$,

  * •

the operator $\Phi_{\mathcal{A}}$ is an algebra homomorphism,

  * •

$\Phi_{\mathcal{A}}$ is the unique $\mathcal{B}^{n}$-calculus for
${\mathcal{A}}$.

The first two properties establish that $\Phi_{\mathcal{A}}$ is a
$\mathcal{B}^{n}$-calculus for ${\mathcal{A}}$. For $n=1$ all three properties
have been proved, the first two in [6, Theorem 4.4], and the third in [7,
Theorem 6.6]. For $n=2$, they have been proved in [21]. The proofs in those
papers can probably be adapted to cover the same properties for arbitrary
$n\in{\mathbb{N}}$, but we will present more direct proofs of all three
properties even for $n=1$.

We proceed with three lemmas which combine to prove the three properties, but
the three proofs are inter-related. We start by considering two non-zero
elementary functions $f$ and $g$ in $\mathcal{B}^{n}$. We denote their
supports by $\Omega_{f}$ and $\Omega_{g}$ with cardinalities $k$ and $m$,
respectively. Thus $f$ and $g$ belong to the spaces
$\mathcal{B}_{0}^{\Omega_{f}}$ and $\mathcal{B}_{0}^{\Omega_{g}}$,
respectively. Unlike the case when $n=1$, the two supports may be different.
As it turns out, this does not cause a serious complication.

######  Lemma 5.1.

Let $\mathcal{A}=(A_{1},\dots,A_{n})$ be an $n$-tuple of commuting operators
which satisfy the full (GSF) condition on a Banach space $X$. Let $f$ and $g$
be elementary functions in $\mathcal{B}^{n}$, and assume that
$g({\mathcal{A}}+{\bm{t}})\in L(X)$ for ${\bm{t}}=(t,\dots,t)$, $t\geq 0$.
Then $fg$ is an elementary function in $\mathcal{B}^{n}$ and

(5.1) $(fg)(\mathcal{A})=f(\mathcal{A})g(\mathcal{A}).$

In particular,

$(f\,\bm{r}_{\mathbf{z}})({\mathcal{A}})=f({\mathcal{A}})\mathbf{r}_{\mathbf{z}}({\mathcal{A}}),\qquad\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

###### Proof.

If $f$ or $g$ is the zero function, the result holds. So we assume that $f$
and $g$ are not zero, and their supports have cardinalities $k$ and $m$,
respectively. It is clear that $fg$ is an elementary function in
$\mathcal{B}^{n}$, and its support is $\Omega:=\Omega_{f}\cup\Omega_{g}$.
Consequently we can also assume that $\Omega=I_{n}$.

Let ${\bm{t}}=(t,\dots,t)\in(0,\infty)^{n}$,
$f_{{\bm{t}}}(\mathbf{z})=f(\mathbf{z}+{\bm{t}})$ and
$g_{{\bm{t}}}(\mathbf{z})=g(\mathbf{z}+{\bm{t}})$. Applying (4.3) to $f$ and
$g$, we obtain, for $\mathbf{z}\in{\mathbb{C}}_{+}^{n}$,

$\displaystyle(f_{{\bm{t}}}g_{{\bm{t}}})(\mathbf{z})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{m}}\int_{{\mathbb{C}}_{+}^{k}}\left[\mathcal{K}_{k}(\mathbf{z}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\mathbf{z}_{\Omega_{g}}+{{\bm{t}}}_{\Omega_{g}},\overline{\bm{\nu}})\right](\mathcal{D}_{\Omega_{f}}f)(\bm{\lambda})(\mathcal{D}_{\Omega_{g}}g)(\bm{\nu})\,dV_{k}(\bm{\lambda})\,dV_{m}(\bm{\nu}).$

Differentiation through the integral (which is easily justified) gives

$\displaystyle\mathcal{D}_{n}(f_{{\bm{t}}}g_{{\bm{t}}})(\mathbf{z})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{m}}\int_{{\mathbb{C}}_{+}^{k}}\mathcal{D}_{n}\left[\mathcal{K}_{k}(\mathbf{z}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\mathbf{z}_{\Omega_{g}}+{\bm{t}}_{\Omega_{g}},\overline{\bm{\nu}})\right]$
$\displaystyle\hbox{}\hskip
90.0pt\times(\mathcal{D}_{\Omega_{f}}f)(\bm{\lambda})(\mathcal{D}_{\Omega_{g}}g)(\bm{\nu})\,dV_{k}(\bm{\lambda})\,dV_{m}(\bm{\nu}).$

Combining (4.5), Lemma 3.5, and Fubini’s theorem (see the Appendix for
justification), we infer that

(5.2) $\displaystyle\hbox{}\hskip
15.0pt\langle(f_{{\bm{t}}}g_{{\bm{t}}})(\mathcal{A})x,x^{*}\rangle$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{n}}\langle\mathcal{K}_{n}(\mathcal{A},\overline{\mathbf{z}})x,x^{*}\rangle\int_{{\mathbb{C}}_{+}^{m}}\int_{{\mathbb{C}}_{+}^{k}}\mathcal{D}_{n}\left(\mathcal{K}_{k}(\mathbf{z}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\mathbf{z}_{\Omega_{g}}+{\bm{t}}_{\Omega_{g}},\overline{\bm{\nu}})\right)$
$\displaystyle\hskip
130.0pt\times(\mathcal{D}_{\Omega_{f}}f)(\bm{\lambda})(\mathcal{D}_{\Omega_{g}}g)(\bm{\nu})\,\,dV_{k}(\bm{\lambda})\,dV_{m}(\bm{\nu})\,dV_{n}(\mathbf{z})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{m}}\int_{{\mathbb{C}}_{+}^{k}}(\mathcal{D}_{\Omega_{f}}f)(\bm{\lambda})(\mathcal{D}_{\Omega_{g}}g)(\bm{\nu})\,R_{{\bm{t}}}(\bm{\lambda},\bm{\nu})\,dV_{k}(\bm{\lambda})\,dV_{m}(\bm{\nu}),$

where

$R_{t}({\bm{\lambda}},\bm{\nu})=\int_{{\mathbb{C}}_{+}^{n}}\langle\mathcal{K}_{n}(\mathcal{A},\overline{\mathbf{z}})x,x^{*}\rangle\mathcal{D}_{n}\left(\mathcal{K}_{k}(\mathbf{z}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\mathbf{z}_{\Omega_{g}}+{\bm{t}}_{\Omega_{g}},\overline{\bm{\nu}})\right)dV_{n}(\mathbf{z}).$

For all $\bm{\lambda}\in{\mathbb{C}}^{k}_{+}$ and
$\bm{\nu}\in{\mathbb{C}}_{+}^{m}$,

$\mathcal{K}_{k}(\cdot+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\in\mathcal{LM}^{k}\cap\mathcal{B}_{0}^{k},\qquad\mathcal{K}_{m}(\cdot+{\bm{t}}_{\Omega_{g}},\overline{\bm{\nu}})\in\mathcal{LM}^{m}\cap\mathcal{B}_{0}^{m},$

as functions of $\mathbf{z}_{\Omega_{f}}$ and $\mathbf{z}_{\Omega_{g}}$,
respectively. Both these functions, and their product, are in
$\mathcal{LM}^{n}$. Using (4.5) and the fact that the Hille-Phillips calculus
is an algebra homomorphism, we see that

(5.3) $\displaystyle R_{{\bm{t}}}(\bm{\lambda},\bm{\nu})$
$\displaystyle=\langle[\mathcal{K}_{k}(\cdot+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\cdot+{\bm{t}}_{\Omega_{g}},\overline{\bm{\nu}})](\mathcal{A})x,x^{*}\rangle$
$\displaystyle=\langle\mathcal{K}_{k}(\mathcal{A}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\mathcal{A}_{\Omega_{g}}+{\bm{t}}_{\Omega_{g}},\overline{\bm{\nu}})x,x^{*}\rangle.$

Thus, by (5.2), (4.5), and Lemma 4.6,

(5.4) $\displaystyle\hbox{}\hskip
10.0pt\langle(f_{{\bm{t}}}g_{{\bm{t}}})(\mathcal{A})x,x^{*}\rangle$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{m}}\int_{{\mathbb{C}}_{+}^{k}}(\mathcal{D}_{\Omega_{f}}f)(\bm{\lambda})(\mathcal{D}_{\Omega_{g}}g)(\bm{\nu})$
$\displaystyle\hbox{}\hskip
70.0pt\langle\mathcal{K}_{k}(\mathcal{A}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\mathcal{A}_{\Omega_{g}}+{\bm{t}}_{\Omega_{g}},\overline{\bm{\nu}})x,x^{*}\rangle\,dV_{k}(\bm{\lambda})\,dV_{m}(\bm{\nu})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{k}}\langle(\mathcal{D}_{\Omega_{f}}f)({\bm{\lambda}})\langle
g({\mathcal{A}}+{\bm{t}}){\mathcal{K}}_{k}({\mathcal{A}}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})x,x^{*}\rangle\,dV_{k}({\bm{\lambda}})$
$\displaystyle=\int_{{\mathbb{C}}_{+}^{k}}\langle(\mathcal{D}_{\Omega_{f}}f)({\bm{\lambda}})\langle{\mathcal{K}}_{k}({\mathcal{A}}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})g({\mathcal{A}}+{\bm{t}})x,x^{*}\rangle\,dV_{k}({\bm{\lambda}})$
$\displaystyle=\langle
f(\mathcal{A}+{\bm{t}})g(\mathcal{A}+{\bm{t}})x,x^{*}\rangle.$

Hence, for all $t>0$,

(5.5)
$(f_{{\bm{t}}}g_{{\bm{t}}})(\mathcal{A})=f_{{\bm{t}}}(\mathcal{A})g_{\bm{{\bm{t}}}}(\mathcal{A}).$

Letting $t\to 0$ in (5.5) and using the strong continuity of the shifts on
$\mathcal{B}^{n}$, we obtain the assertion (5.1). ∎

Next we prove that for “smoothed” functions $f\in\mathcal{B}_{0}^{n}$ the
reproducing formula (3.3) can be interpreted as a $\mathcal{B}_{0}^{n}$-valued
Bochner integral. Here $\mathbf{1}=(1,\dots,1)\in{\mathbb{R}}^{n}$, and
$T({\bm{t}})$ is defined in (3.5).

######  Lemma 5.2.

Let $f\in\mathcal{B}_{0}^{n}$ and ${\bm{t}}=(t,\dots,t)\in(0,\infty)^{n}$.
Define $F_{\bm{t}}:{\mathbb{C}}_{+}^{n}\to\mathcal{B}_{0}^{n}$ by

$F_{\bm{t}}({\bm{\lambda}})=\big{(}\mathcal{D}_{n}\left(f\mathbf{r}_{{{\bm{1}}}}^{2}\right)\\!\\!\big{)}({\bm{\lambda}})\,\mathbf{r}_{{\bm{t}}+\overline{\bm{\lambda}}}^{2},\qquad{\bm{\lambda}}\in{\mathbb{C}}_{+}^{n}.$

Then $F_{\bm{t}}$ is continuous and Bochner-integrable with respect to the
measure $dV_{n}({\bm{\lambda}})$. Moreover,

(5.6)
$\left(-\frac{2}{\pi}\right)^{n}\int_{{\mathbb{C}}_{+}^{n}}F_{t}({\bm{\lambda}})\,dV_{n}({\bm{\lambda}})=f_{\bm{t}}\mathbf{r}_{{{\bm{1}}}+{\bm{t}}}^{2}.$

###### Proof.

The function
${\bm{\lambda}}\mapsto\mathbf{r}_{{\bm{t}}+\overline{{\bm{\lambda}}}}^{2}$ is
continuous from ${\mathbb{C}}_{+}^{n}$ to $\mathcal{B}_{0}^{n}$, and hence
$F_{\bm{t}}$ is continuous. If $\lambda_{j}=\alpha_{j}+i\beta_{j}$, then

$\|\mathbf{r}_{{\bm{t}}+\overline{\bm{\lambda}}}^{2}\|_{\mathcal{B}_{0}^{n}}=\prod_{j=1}^{n}\frac{1}{(\alpha_{j}+t)^{2}}.$

Moreover, $\big{(}\mathcal{D}_{n}(fr_{1}^{2})\big{)}({\bm{\lambda}})$ is the
sum of $2^{n}$ terms of the following form, indexed by
$\Omega\in\mathcal{P}_{n}$, where $k=|\Omega|$:

$J_{n,\Omega}({\bm{\lambda}}):=(-2)^{n-k}(\mathcal{D}_{\Omega}f)({\bm{\lambda}})\prod_{j\in\Omega}r_{1}(\lambda_{j})^{2}\prod_{j\in\Omega^{c}}r_{1}(\lambda_{j})^{3},$

and, by a standard estimate,

$|(\mathcal{D}_{\Omega}f)({\bm{\lambda}})|\leq\frac{\|f\|_{\infty}}{2^{k}\prod_{j\in\Omega}\alpha_{j}}.$

It follows that

$\displaystyle\hskip
0.0pt\int_{{\mathbb{C}}_{+}^{n}}|J_{n,\Omega}({\bm{\lambda}})|\,\|\mathbf{r}^{2}_{{\bm{t}}+\overline{{\bm{\lambda}}}}\|_{\mathcal{B}_{0}^{n}}\,dV_{n}({\bm{\lambda}})$
$\displaystyle\leq
2^{n-2k}\|f\|_{\infty}\int_{{\mathbb{C}}_{+}^{n}}\prod_{j\in\Omega}\frac{\alpha_{j}}{\alpha_{j}(\alpha_{j}+t)^{2}|1+\lambda_{j}|^{2}}\prod_{j\in\Omega^{c}}\frac{\alpha_{j}}{(\alpha_{j}+t)^{2}|1+\lambda_{j}|^{3}}\,dS_{n}({\bm{\alpha}},{\bm{\beta}})$
$\displaystyle\leq
2^{n-2k}\pi^{n}\|f\|_{\infty}\left(\int_{0}^{\infty}\frac{d\alpha}{(\alpha+t)^{2}}\right)^{n}=\frac{(2\pi)^{n}}{4^{k}t^{n}}\|f\|_{\infty}.$

Hence

$\int_{{\mathbb{C}}_{+}^{n}}|\mathcal{D}_{n}(f\mathbf{r}_{1})({\bm{\lambda}})|\,\|\mathbf{r}^{2}_{{\bm{t}}+\overline{\bm{\lambda}}}\|_{t_{0}^{n}}\,dV_{n}({\bm{\lambda}})\leq\frac{(2\pi)^{n}}{t^{n}}\sum_{k=0}^{n}{n\choose
k}4^{-k}\|f\|_{\infty}=\left(\frac{5\pi}{2t}\right)^{n}\|f\|_{\infty}.$

Thus,
$\int_{{\mathbb{C}}_{+}^{n}}\|F_{\bm{t}}({\bm{\lambda}})\|_{\mathcal{B}_{0}^{n}}\,dV_{n}({\bm{\lambda}})<\infty$,
so the Bochner integral of $F_{t}$ exists. Since point evaluations are
continuous linear functionals on $\mathcal{B}_{0}^{n}$, the function on the
left-hand side of (5.6) maps $\mathbf{z}\in{\mathbb{C}}_{+}^{n}$ to

$\left(-\frac{2}{\pi}\right)^{n}\int_{{\mathbb{C}}_{+}^{n}}F_{\bm{t}}({\bm{\lambda}})(\mathbf{z})\,dV_{n}({\bm{\lambda}})=\int_{{\mathbb{C}}_{+}^{n}}{\mathcal{K}}_{n}(\mathbf{z}+{\bm{t}},\overline{\bm{\lambda}})\mathcal{D}_{n}(f\mathbf{r}_{{\bm{1}}}^{2})\,dV_{n}({\bm{\lambda}})=(f\mathbf{r}_{{\bm{1}}}^{2})(\mathbf{z}+{\bm{t}}),$

using Lemma 3.5. Hence (5.6) holds. ∎

Next, we apply the mapping $\Phi_{\mathcal{A}}$ to both sides of (5.6), and
then deduce the three properties of $\Phi_{\mathcal{A}}$ simultaneously. We
say that

$\Upsilon:\,\mathcal{B}_{0}^{n}\mapsto L(X,X^{**})$

is a (bounded) “semi-homomorphism” if $\Upsilon$ is a bounded linear map
satisfying

(5.7)
$\Upsilon(f\cdot\mathbf{r}_{\mathbf{z}})=\Upsilon(f)(\mathbf{z}+{\mathcal{A}})^{-1},\qquad
f\in\mathcal{B}_{0}^{n},\;\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

Lemma 5.1 shows that $\Phi_{A}$ is a semi-homomorphism.

######  Lemma 5.3.

Let ${\mathcal{A}}=(A_{1},\dots,A_{n})$ be an $n$-tuple of commuting operators
on $X$ such that $\sigma(A_{j})\subseteq\overline{{\mathbb{C}}}_{+}$ for each
$j\in I_{n}$. Let

$\Upsilon:\,\mathcal{B}_{0}^{n}\mapsto\mathcal{L}(X,X^{**})$

be a semi-homomorphism such that
$\Upsilon(\mathbf{r}_{\mathbf{z}})=\mathbf{r}_{\mathbf{z}}({\mathcal{A}}),\mathbf{z}\in{\mathbb{C}}_{+}^{n}$.
Then, for all $f\in\mathcal{B}_{0}^{n}$,

(5.8) $\Upsilon(f)\mathbf{r}_{{\bm{1}}}({\mathcal{A}})^{2}=\lim_{t\to
0}\,Q_{t}(f;{\mathcal{A}}),$

in operator-norm, where $Q_{t}(f;{\mathcal{A}})\in{L}(X)$ is defined for $t>0$
by

(5.9)
$Q_{t}(f;{\mathcal{A}}):=\left(-\frac{2}{\pi}\right)^{n}\int_{{\mathbb{C}}_{+}^{n}}(\mathcal{D}_{n}(f\mathbf{r}_{{\bm{1}}}^{2}))({\bm{\lambda}})({\mathcal{A}}+{\bm{t}}+\overline{\bm{\lambda}})^{-2}\,dV_{n}({\bm{\lambda}}).$

Consequently, $\Upsilon$ is uniquely determined by its values on
$\\{\mathbf{r}_{\mathbf{z}}:\mathbf{z}\in{\mathbb{C}}_{+}^{n}\\}$ and

(5.10) $\Upsilon(f)\in{L}(X),\quad f\in\mathcal{B}_{0}^{n}.$

###### Proof.

Using (5.7), Lemma 5.2, and Lemma 4.6, and applying $\Upsilon$ to both sides
of (5.6), we have

(5.11)
$\displaystyle\Upsilon(f_{\bm{t}})\mathbf{r}_{{{\bm{1}}}+{\bm{t}}}({\mathcal{A}})^{2}=\Upsilon(f_{\bm{t}}\cdot\mathbf{r}_{{{\bm{1}}}+{\bm{t}}}^{2})=Q_{t}(f;{\mathcal{A}}),\qquad
t>0,$

where $Q_{t}(f;A)$ is given by (5.9), and the integral converges in the norm
of $L(X)$. By the continuity of shifts on $\mathcal{B}_{0}^{n}$, as in Lemma
3.6,

$\lim_{t\to 0}\,\|\Upsilon(f_{\bm{t}})-\Upsilon(f)\|_{{L}(X,X^{**})}=0,$

and, by the resolvent identity,

$\lim_{t\to
0}\,\|\mathbf{r}_{{{\bm{1}}}+{\bm{t}}}({\mathcal{A}})-\mathbf{r}_{{{\bm{1}}}}({\mathcal{A}})\|_{{L}(X)}=0.$

Thus, (5.11) implies (5.8) and (5.10).

Since ${\rm dom}\,(A_{j})$ is dense in $X$, $r_{1}(A_{j})^{2}$ has dense range
for $j\in I_{n}$, and then $\mathbf{r}_{{\bm{1}}}({\mathcal{A}})^{2}$ has
dense range. The uniqueness of $\Upsilon$ follows from this and (5.8) and
(5.9). Since $X$ is closed in $X^{**}$, the assertion (5.10) holds. ∎

Now we can conclude that $\Phi_{A}$, as defined in (4.5) and (4.6), is a
$\mathcal{B}^{n}$-calculus for ${\mathcal{A}}$, as defined in Definition 4.1.

######  Theorem 5.4.

Let

$\mathcal{A}=(A_{1},\dots,A_{n})$

satisfy the full (GSF) condition on a Banach space $X$. Then $\Phi_{A}$ is a
$\mathcal{B}^{n}$-calculus for ${\mathcal{A}}$, and it is unique.

###### Proof.

Lemma 5.3 shows that $\Phi_{\mathcal{A}}$ maps $\mathcal{B}_{0}^{n}$ into
$L(X)$, and the same holds for the maps on $\mathcal{B}_{\Omega,0}^{n}$. By
Lemma 5.1, $\Phi_{\mathcal{A}}$ is multiplicative on elementary functions and
hence on $\mathcal{B}^{n}$. Then $\Phi_{\mathcal{A}}$ maps $\mathcal{B}^{n}$
into $L(X)$, and is multiplicative on all functions in $\mathcal{B}^{n}$. In
addition, $\Phi_{A}$ maps the function $\mathbf{z}\mapsto(z_{j}+\lambda)^{-1}$
to the operator $(A_{j}+\lambda)^{-1}$ for $\lambda\in{\mathbb{C}}_{+},\,j\in
I_{n}$, so $\Phi_{\mathcal{A}}$ is a $\mathcal{B}^{n}$-calculus for
${\mathcal{A}}$.

Let $f\in\mathcal{B}^{n}$. Then $f\mathbf{r}_{{\bm{1}}}\in\mathcal{B}^{n}_{0}$
and

$\Phi_{\mathcal{A}}(f)\mathbf{r}_{{\bm{1}}}({\mathcal{A}})=\Phi_{A}(f\mathbf{r}_{{\bm{1}}}).$

Since $\Phi_{A}(f\mathbf{r}_{{\bm{1}}})$ is uniquely determined (Lemma 5.3)
and the range of $\mathbf{r}_{{\bm{1}}}({\mathcal{A}})$ is dense in $X$, it
follows that $\Phi_{A}(f)$ is uniquely determined. ∎

##  6\. Mergers and sums

In this section, we consider two situations where there might be questions of
compatibility between different $\mathcal{B}$-calculi.

### Mergers

In some cases, two or more of the operators in an $n$-tuple ${\mathcal{A}}$
may coincide. We may merge some or all of the repeated operators to form an
$m$-tuple $\widetilde{\mathcal{A}}=(\widetilde{A}_{k})_{k\in I_{m}}$, where
there is a surjective function $\pi$ from $I_{n}$ to $I_{m}$ such that
$A_{j}=\widetilde{A}_{\pi(j)}$ for all $j\in I_{n}$. If ${\mathcal{A}}$
satisfies (GSF), then it is trivial that $\widetilde{\mathcal{A}}$ also
satisfies the full (GSF), because (GSF${}_{\widetilde{\Omega}}$) for
$\widetilde{\mathcal{A}}$ coincides precisely with (GSFΩ) for ${\mathcal{A}}$
for some $\Omega\in\mathcal{P}_{n}$ where $\pi$ is a bijection from
$\widetilde{\Omega}$ to $\Omega$. We show in this section that the
$\mathcal{B}$-calculi for $\widetilde{\mathcal{A}}$ and ${\mathcal{A}}$ are
compatible, thus establishing that mergers of this type within the
$\mathcal{B}$-calculus are valid. A similar result in a different context has
been obtained in [2, Proposition 4.1].

Let $f\in\mathcal{B}^{n}$, and define

(6.1) $(\Upsilon
f)(\mathbf{w}):=f(\mathbf{z}),\quad\mathbf{w}=(w_{1},\dots,w_{m})\in{\mathbb{C}}_{+}^{m},\;z_{j}=w_{\pi(j)}.$

######  Proposition 6.1.

Let ${\mathcal{A}}$, $\widetilde{\mathcal{A}}$ and $\Upsilon$ be as above, and
assume that ${\mathcal{A}}$ satisfies the full (GSF). Let
$f\in\mathcal{B}^{n}$. Then $\Upsilon(f)\in\mathcal{B}^{m}$, and $(\Upsilon
f)(\widetilde{\mathcal{A}})=f({\mathcal{A}})$ in the setting of their
respective $\mathcal{B}$-calculi.

###### Proof.

To see that $\Upsilon f\in\mathcal{B}^{m}$, we first consider the case when
$m=n-1$, and $\pi(j)=\min(j,n-1)$. Let $g=\Upsilon f$, and
$\Omega\in\mathcal{P}_{n-1}$.

Firstly, assume that $z_{n-1}\notin\Omega$. Then
$(\mathcal{D}_{\Omega}g)(\mathbf{z})=(\mathcal{D}_{\Omega}f)(\mathbf{z},z_{n-1})$,
for $\mathbf{z}\in{\mathbb{C}}_{+}^{n-1}$, and
$(H_{\Omega}g)({\bm{\alpha}}_{\Omega})\leq(H_{\Omega}f)({\bm{\alpha}}_{\Omega})$,
so $H_{\Omega}g$ is integrable. Secondly, assume that $n-1\in\Omega$. Let
$\tilde{\Omega}=(\Omega\setminus\\{n-1\\})\cup\\{n\\}$. Then

$(\mathcal{D}_{\Omega}g)(\mathbf{z})=(\mathcal{D}_{\Omega}f)(\mathbf{z},z_{n-1})+(\mathcal{D}_{\tilde{\Omega}}f)(\mathbf{z},z_{n-1}).$

Hence

$(H_{\Omega}g)({\bm{\alpha}}_{\Omega})\leq(H_{\Omega}f)({\bm{\alpha}}_{\Omega})+(H_{\tilde{\Omega}}f)({\bm{\alpha}}_{\tilde{\Omega}}).$

It follows that $H_{\Omega}g$ is integrable. Thus $g\in\mathcal{B}^{n-1}$, and
the map $\Upsilon:\mathcal{B}^{n}\to\mathcal{B}^{n-1}$ is bounded. By symmetry
of the variables, this holds for any simple merger of just one pair of
variables. The general case follows by carrying out $n-m$ simple mergers.

It is clear that $\Upsilon$ is a bounded algebra homomorphism from
$\mathcal{B}^{n}$ to $\mathcal{B}^{m}$. Let
$\Upsilon(f)=\Phi_{\widetilde{\mathcal{A}}}(\Upsilon f)$, so
$\Upsilon:\mathcal{B}^{n}\to L(X)$ is a bounded algebra homomorphism. Let
$r_{\lambda,j}(\mathbf{z})=(z_{j}+\lambda)^{-1}$ where
$\mathbf{z}\in{\mathbb{C}}_{+}^{n}$. Then, for $j\in I_{n}$ and
$\lambda\in{\mathbb{C}}_{+}$,

$\Upsilon r_{\lambda,j}=r_{\lambda,\pi(j)},$

and

$\Psi(r_{\lambda,j})=(\widetilde{A}_{\pi(j)}+\lambda)^{-1}=(A_{j}+\lambda)^{-1}.$

It follows from the uniqueness statement in Theorem 5.4 that
$\Upsilon=\Phi_{\mathcal{A}}$, as required. ∎

### Sums of operators

It was observed in [6, Lemma 2.6(5)] that if $f\in\mathcal{B}^{1}$, $b>0$ and
$g(z)=f(bz)$, then $g\in\mathcal{B}^{1}$ and
$\|g\|_{\mathcal{B}^{1}}=\|f\|_{\mathcal{B}^{1}}$. Moreover, if $A$ satisfies
the (GSF) condition, then $g(A)=f(bA)$. This can be extended to the case of
several variables, and the proofs are very simple changes of variables. Such
scalings can be combined with mergers as above. For example, suppose that
$f\in\mathcal{B}^{2}$ and $g(z)=(b_{1}z,b_{2}z)$, where $b_{1},b_{2}>0$, and
$A$ satisfies the (GSF) condition, then $g(A)=f(b_{1}A,b_{2}A)$.

Here we discuss the case when the operators in ${\mathcal{A}}$ may be added
together, in the way that the variables $z_{j}$ were added in Proposition 3.8.
We present only the case all the operators are added, but more general cases
can be considered by the same techniques.

Let ${\mathcal{A}}$ be an $n$-tuple of commuting operators which satisfy the
full (GSF) condition, $f\in\mathcal{B}^{1}$ and
$f^{[n]}(\mathbf{z})=f(z_{1}+\dots+z_{n})$ for
$\mathbf{z}\in{\mathbb{C}}_{+}^{n}$. By Proposition 3.8,
$f^{[n]}\in\mathcal{B}^{n}$. Thus $f^{[n]}({\mathcal{A}})$ is defined in the
$\mathcal{B}^{n}$-calculus, but $f(A_{1}+\dots+A_{n})$ is not defined in the
$\mathcal{B}^{1}$-calculus (except in rare cases), because the operator
$G_{0}:=A_{1}+\dots+A_{n}$, with domain
$\operatorname{dom}(G_{0})=\operatorname{dom}(A_{1})\cap\dots\cap\operatorname{dom}(A_{n})$,
is not closed. However $\operatorname{dom}(G_{0})$ is dense in $X$ and the
individual operators $-A_{j}$ generate commuting bounded $C_{0}$-semigroups
$(e^{-tA_{j}})_{t\geq 0}$, and the operators
$T_{\mathcal{A}}(t):=\prod_{j=1}^{n}e^{-tA_{j}}$ form a bounded
$C_{0}$-semigroup whose negative generator $G$ is the closure of $G_{0}$ (see
[13, Theorem 1.9]).

The following result was proved for $n=2$ in [21, Lemma 6.2.3].

######  Proposition 6.2.

Let ${\mathcal{A}}$ be an $n$-tuple of operators which satisfy the full (GSF),
and let $G$ be the closure of $A_{1}+\dots+A_{n}$. Then $G$ satisfies (GSF1)
and $f(G)=f^{[n]}({\mathcal{A}})$ for all $f\in\mathcal{B}^{1}$.

###### Proof.

The mapping $f\mapsto f^{[n]}({\mathcal{A}})$ is a bounded algebra
homomorphism from $\mathcal{B}^{1}$ to $L(X)$. It suffices to prove that
$r_{\lambda}^{[n]}({\mathcal{A}})=r_{\lambda}(G)$ for
$\lambda\in{\mathbb{C}}_{+}$. Then $G$ has a $\mathcal{B}$-calculus, so $G$
satisfies (GSF1), and uniqueness of the $\mathcal{B}^{1}$-calculus implies
that $f(G)=f^{[n]}({\mathcal{A}})$.

Let $\mu$ be the measure on ${\mathbb{C}}_{+}^{n}$ given by
$\mu(E)=\int_{\tilde{E}}e^{-\lambda t}\,dt$, where
$\tilde{E}=\\{t\in{\mathbb{C}}_{+}:(t,t,\dots,t)\in E\\}$. Then
$r_{\lambda}^{[n]}$ is the Laplace transform of the measure $\mu$. Since the
$\mathcal{B}^{n}$-calculus agrees with the HP-calculus on $\mathcal{LM}^{n}$,

$\displaystyle r_{\lambda}^{[n]}({\mathcal{A}})x$
$\displaystyle=\int_{{\mathbb{R}}_{+}^{n}}\bigg{(}\prod_{j=1}^{n}e^{-t_{j}A_{j}}\bigg{)}x\,d\mu({\bm{t}})=\int_{{\mathbb{R}}_{+}}e^{-\lambda
t}e^{-tG}x\,dt=r_{\lambda}(G)x.\qed$

##  7\. Spectral mapping properties

Let ${\mathcal{A}}$ be an $n$-tuple of commuting operators on a Banach space
$X$, satisfying the full (GSF) condition, so ${\mathcal{A}}$ has a
$\mathcal{B}^{n}$-calculus. We will use the Shilov joint spectrum of
${\mathcal{A}}$ as in [26, Definition 9] based on [18, Theorem 16.3.1].

Let ${\mathfrak{A}}$ be the (commutative) Banach subalgebra of $L(X)$
generated by all operators of the form $f({\mathcal{A}})$ for
$f\in\mathcal{B}^{n}$ and all their resolvents. The spectrum of
$f({\mathcal{A}})$ in ${\mathfrak{A}}$ coincides with the spectrum in $L(X)$,
and it will be denoted by $\sigma(f({\mathcal{A}}))$. For each $j\in I_{n}$,
the Gelfand spectrum of ${\mathfrak{A}}$ splits into two subsets
${\mathfrak{M}}_{j}$ and ${\mathfrak{N}}_{j}$ and there is a continuous
function $\eta_{j}$ on ${\mathfrak{M}}_{j}$ such that, for all
$z\in{\mathbb{C}}_{+}$,

$\chi((A_{j}+z)^{-1})=\begin{cases}(\eta_{j}(\chi)+z)^{-1}&\text{if
$\chi\in{\mathfrak{M}}_{j}$},\\\ 0&\text{if
$\chi\in{\mathfrak{N}}_{j}$}.\end{cases}$

Then $\sigma(A_{j})=\eta_{j}({\mathfrak{M}}_{j})$. The _Shilov joint spectrum_
$\sigma({\mathcal{A}})$ of ${\mathcal{A}}$ is defined to be

$\sigma({\mathcal{A}})=\left\\{(\eta_{1}(\chi),\dots,\eta_{n}(\chi)):\chi\in\cap_{j\in
I_{n}}{\mathfrak{M}}_{j}\right\\}.$

We have the following spectral inclusion theorem.

######  Theorem 7.1.

Let ${\mathcal{A}}$ be an $n$-tuple of operators which satisfy the full (GSF)
condition, and let $f\in\mathcal{B}^{n}$ and
${\bm{\lambda}}\in\sigma({\mathcal{A}})$. Then
$f({\bm{\lambda}})\in\sigma(f({\mathcal{A}}))$.

###### Proof.

First we will assume that the resolvents of $A_{j}$ are bounded on the left
half-plane. It follows from this assumption that there exists $C$ such that,
for all $z\in{\mathbb{C}}_{+}$,

(7.1) $\|(A_{j}+z)^{-2}(A_{j}+1)^{-2}\|\leq C(1+|z-1|)^{-2},\quad j\in I_{n}.$

Let $f\in\mathcal{B}_{0}^{n}$. Then (4.5) and (7.1) can be used to obtain

(7.2)
$f({\mathcal{A}})r_{{\bm{1}}}({\mathcal{A}})^{2}=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{K}_{n}({\mathcal{A}},\overline{{\bm{\nu}}})r_{{\bm{1}}}({\mathcal{A}})^{2}\,(\mathcal{D}_{n}f)({\bm{\nu}})\,dV_{n}({\bm{\nu}}),$

where the integral converges in operator norm.

Let ${\bm{\lambda}}\in\sigma({\mathcal{A}})$. There is a character $\chi$ of
${\mathfrak{A}}$ such that $\chi\in{\mathfrak{M}}_{j}$ and
$\chi((A_{j}+1)^{-1})=(\lambda_{j}+1)^{-1}$ for all $j\in I_{n}$. Then
$\chi(r_{{{\bm{1}}}}({\mathcal{A}}))=r_{{\bm{1}}}({\bm{\lambda}})$ and
$\chi(\mathcal{K}_{n}({\mathcal{A}},{\bm{\nu}}))=\mathcal{K}_{n}({\bm{\lambda}},{\bm{\nu}})$.
Applying $\chi$ to (7.2), we obtain

$\chi(f({\mathcal{A}}))r_{{\bm{1}}}({\bm{\lambda}})^{2}=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{K}_{n}({\bm{\lambda}},\overline{{\bm{\nu}}})r_{{\bm{1}}}({\bm{\lambda}})^{2}\,(\mathcal{D}_{n}f)({\bm{\nu}})\,dV_{n}({\bm{\nu}})=f({\bm{\lambda}})r_{{\bm{1}}}({\bm{\lambda}})^{2}.$

Thus $f({\bm{\lambda}})=\chi(f({\mathcal{A}}))$.

For a general $f\in\mathcal{B}^{n}$, one may consider the terms $f_{\Omega,0}$
in the elementary decomposition of $f$, using the same character $\chi$,
adapting the formulas above accordingly to show that
$f_{\Omega,0}({\bm{\lambda}}_{\Omega})=\chi(f_{\Omega,0}({\mathcal{A}}_{\Omega}))$,
and then summing over $\Omega\in\mathcal{P}_{n}$ to deduce that
$f({\bm{\lambda}})=\chi(f({\mathcal{A}}))$.

Now we cease to assume that the resolvents of $A_{j}$ are bounded on the left
half-plane. Instead, we can consider the operators $A_{j}+\varepsilon$, where
$\varepsilon>0$, and apply the result above. The same $\chi$ as above may be
used for sufficently small $\varepsilon>0$ to show that
$f({\bm{\lambda}}+\bm{\varepsilon})=\chi(f({\mathcal{A}}+\bm{\varepsilon}))$.
Letting $\varepsilon\to 0$ and using Lemmas 3.6 and 4.6, it follows that
$f({\bm{\lambda}})=\chi(f({\mathcal{A}}))\in\sigma(f({\mathcal{A}}))$. ∎

There are other results on spectral mappings for the
$\mathcal{B}^{n}$-calculus, as follows.

######  Proposition 7.2.

Let ${\mathcal{A}}$ be an $n$-tuple of operators on a Banach space $X$,
satisfying the full (GSF) condition, and let $f\in\mathcal{B}^{n}$. The
following results hold.

  1. 1.

If $x\in X$ is an eigenvector of $A_{j}$ with eigenvalue $\lambda_{j}$ for
each $j\in I_{n}$, then $x$ is an eigenvector for $f({\mathcal{A}})$ with
eigenvalue $f({\bm{\lambda}})$.

  2. 2.

If $(x_{k})_{k\in{\mathbb{N}}}$ are unit vectors in $X$ forming an approximate
eigenvector of $A_{j}$ with approximate eigenvalue $\lambda_{j}$ for each
$j\in I_{n}$, then $f({\bm{\lambda}})$ is an approximate eigenvalue of $f(A)$.

###### Proof.

The first statement can be seen very easily from the definition of $f(A)$ in
(4.3).

The second statement follows from the first, using an $F$-product as in [27].
For this, let

$\displaystyle\ell^{\infty}_{T}(X)$
$\displaystyle=\Big{\\{}(x_{k})_{k\in{\mathbb{N}}}:\sup\|x_{k}\|<\infty,\;\lim_{t\to
0}\sum_{j=1}^{n}\sup_{k\in{\mathbb{N}}}\left\|e^{-tA_{j}}x_{k}-x_{k}\right\|=0\Big{\\}},$
$\displaystyle c_{0}(X)$
$\displaystyle=\big{\\{}(x_{k})_{k\in{\mathbb{N}}}:\lim_{k\to\infty}\|x_{k}\|=0\big{\\}}.$

Then $c_{0}(X)$ is a closed subspace of $\ell^{\infty}_{T}(X)$. Let
$\mathcal{X}_{T}=\ell^{\infty}_{T}(X)/c_{0}(X)$, and
$\pi:\ell^{\infty}_{T}(X)\to\mathcal{X}_{T}$ be the quotient map. For each
$j\in I_{n}$, the $C_{0}$-semigroup $(e^{-tA_{j}})_{t\geq 0}$ induces a
$C_{0}$-semigroup on $\mathcal{X}_{T}$. Let $-Z_{j}$ be the generator, and
$\mathcal{Z}=(Z_{1},\dots,Z_{n}$). The operators $f(\mathcal{Z})$ on $X$
induce operators on $\mathcal{X}_{T}$ which form a $\mathcal{B}^{n}$-calculus
for $\mathcal{Z}$.

There are standard methods to show that (a) an approximate eigenvector
$(x_{k})$ for ${\mathcal{A}}$ is mapped by $\pi$ to an approximate eigenvector
for $\mathcal{Z}$ with eigenvalues ${\bm{\lambda}}$, and (b) ${\bm{\lambda}}$
is an eigenvalue for $\mathcal{Z}$. By the first statement (1),
$f({\bm{\lambda}})$ is an eigenvalue of $f(\mathcal{Z})$. It then follows
directly that ${\bm{\lambda}}$ is an approximate eigenvalue of
$f({\mathcal{A}})$. Details for (a) and (b) in the cases $n=1$ and $n=2$ may
be found in [27, pp. 20, 78] and [21, Propositions 2.3.17, 6.4.4]. ∎

If all the operators $A_{j}$ are sectorial of angle less than $\pi/2$, then
${\mathcal{A}}$ satisfies the full (GSF) condition and the formula (4.5) can
be adapted to

(7.3)
$f({\mathcal{A}})=\int_{{\mathbb{C}}_{+}^{n}}\mathcal{K}_{n}({\mathcal{A}},\overline{{\bm{\nu}}})\,(\mathcal{D}_{n}f)({\bm{\nu}})\,dV_{n}({\bm{\nu}}),$

where the integral is convergent in operator norm.

######  Theorem 7.3.

Let ${\mathcal{A}}$ be an $n$-tuple of sectorial operators of angle less than
$\pi/2$ on a Banach space $X$, and let $f\in\mathcal{B}^{n}$. Then

$\sigma(f({\mathcal{A}}))\subseteq\bigcup_{\Omega\in\mathcal{P}_{n}}f_{\Omega}(\sigma({\mathcal{A}})).$

###### Proof.

Let $\chi$ be a character of ${\mathfrak{A}}$, and $\Omega_{\chi}=\\{j\in
I_{n}:\chi\in{\mathfrak{M}}_{j}\\}$. For $j\in\Omega_{\chi}$, let
$\lambda_{j}$ satisfy $\chi((z+A_{j})^{-1})=(z+\lambda_{j})^{-1}$ for all
$z\in{\mathbb{C}}_{+}$.

Let $\Omega\in\mathcal{P}_{n}$, and $k=|\Omega|$. Assume first that
$\Omega\subseteq\Omega_{\chi}$. Then
${\bm{\lambda}}_{\Omega}\in\sigma({\mathcal{A}}_{\Omega})$, and
$\chi({\mathcal{K}}_{k}({\mathcal{A}}_{\Omega},\overline{{\bm{\nu}}}))={\mathcal{K}}_{k}({\bm{\lambda}}_{\Omega},\overline{{\bm{\nu}}})$.
It follows from (7.3) that

$\chi(f_{\Omega,0}({\mathcal{A}}))=\int_{{\mathbb{C}}_{+}^{k}}\mathcal{K}_{k}({\bm{\lambda}}_{\Omega},\overline{{\bm{\nu}}})\,(\mathcal{D}_{k}f_{\Omega,0})({\bm{\nu}})\,dV_{n}({\bm{\nu}})=f_{\Omega,0}({\bm{\lambda}}_{\Omega}).$

Note that this conclusion is valid even if $\Omega$ is the empty set. Then
$f_{\Omega,0}$ is a scalar.

If $\Omega$ is not contained in $\Omega_{\chi}$, then, for some $j\in\Omega$,
$\chi(r_{z}(A_{j}))=0$ for all $z\in{\mathbb{C}}_{+}$, so
$\chi(\mathcal{K}_{k}({\mathcal{A}},\overline{{\bm{\nu}}}))=0$ and
$\chi(f_{\Omega,0}({\mathcal{A}}))=0=f_{\Omega,0}({\bm{\lambda}})$.

Summing over all $\Omega\in\mathcal{P}_{n}$, we obtain that
$\chi(f({\mathcal{A}}))=f_{\Omega_{\chi}}({\bm{\lambda}})$. ∎

######  Corollary 7.4.

Let ${\mathcal{A}}$ be an $n$-tuple of sectorial operators of angle less than
$\pi/2$ on a Banach space $X$, and let $f\in\mathcal{B}_{0}^{n}$. Then

$\sigma(f({\mathcal{A}}))=f(\sigma({\mathcal{A}})).$

###### Proof.

Theorem 7.1 shows that
$f(\sigma({\mathcal{A}}))\subseteq\sigma(f({\mathcal{A}}))$. For the reverse
inclusion, $f_{\Omega,0}=0$ unless $\Omega=\mathcal{P}_{n}$. Then the proof of
Theorem 7.3 shows that $\chi(f({\mathcal{A}}))=f({\bm{\lambda}})$ for some
${\bm{\lambda}}\in\sigma({\mathcal{A}})$, as required. ∎

##  8\. Estimates for functions and operators

Let $\mathcal{A}=(A_{1},\dots,A_{n})$ be an $n$-tuple satisfying the full
(GSF) condition. In this section, we will use the $\mathcal{B}^{n}$-calculus
constructed in Theorem 5.4 to provide several estimates for $f(\mathcal{A})$
in terms of $\|f\|_{\infty}$ for $f$ belonging to substantial subclasses of
$H^{\infty}(\mathbb{C}^{n}_{+})$. The estimates resemble the situation of a
bounded $H^{\infty}$-calculus and thus could be potentially useful.

We will need a Fourier characterisation of functions in
$H^{\infty}(\mathbb{C}^{n}_{+})$ in terms of their boundary values. If $f\in
H^{\infty}(\mathbb{C}^{n}_{+})$, then
$f^{b}({\bm{\beta}}):=\lim_{{\bm{\alpha}}\to
0+}f({\bm{\alpha}}+i{\bm{\beta}})$ exists for almost all
${\bm{\beta}}\in\mathbb{R}^{n}$ (see [23, Theorem F]). Then $f^{b}\in
L^{\infty}(\mathbb{R}^{n}_{+})$. For $f\in L^{\infty}(\mathbb{R}^{n}_{+}),$
define the spectrum of $f$ as ${\rm sp}(f):={\rm supp}\,(\mathcal{F}^{-1}f),$
where $\mathcal{F}$ stands for the (distributional) Fourier transform of $f$.
Let $H^{\infty}(\mathbb{R}^{n}):=\\{f\in L^{\infty}(\mathbb{R}^{n}):{\rm
sp}\,(f)\subseteq\mathbb{R}^{n}_{+}\\}$. Then $H^{\infty}(\mathbb{C}^{n}_{+})$
is isometrically isomorphic to $H^{\infty}(\mathbb{R}^{n})$ via the mapping
$f\to f^{b},$ and the inverse mapping is given by

(8.1)
$f({\bm{\alpha}}+i{\bm{\beta}})=(P_{{\bm{\beta}}}*f^{b})({\bm{\alpha}})=\int_{\mathbb{R}^{n}}P({\bm{\alpha}}-{\bm{t}},{\bm{\beta}})f^{b}(\mathbf{t})\,d\mathbf{t},$

where ${\bm{\alpha}}\in\mathbb{R}^{n}_{+},\,{\bm{\beta}}\in\mathbb{R}^{n}$,
and

$P_{{\bm{\beta}}}({\bm{\alpha}})=P({\bm{\alpha}},{\bm{\beta}})=\prod_{j=1}^{n}\frac{1}{\pi}\frac{\beta_{j}}{\alpha_{j}^{2}+\beta_{j}^{2}}$

stands for the Poisson kernel. This fact is well-known for $n=1$ (see [17,
Section II.1.5]). For $n>1$ it follows from [23, Theorem 2.7].

For $\tau>0$, let $H^{\infty}([\tau,\infty)^{n}):=\\{f\in
H^{\infty}(\mathbb{C}_{+}^{n}):{\rm
sp}\,(f^{b})\subseteq[\tau,\infty)^{n}\\}$, and

$e_{\tau}(\mathbf{z})=e^{-\tau(z_{1}+\dots+z_{n})},\qquad\mathbf{z}\in\mathbb{C}_{+}^{n}.$

Then

(8.2) $H^{\infty}([\tau,\infty)^{n})=e_{\tau}H^{\infty}(\mathbb{C}^{n}_{+}),$

so that from (8.1), for $f\in H^{\infty}([\tau,\infty]^{n})$,

(8.3) $|f(\mathbf{z})|\leq e^{-\tau(\sum_{i=1}^{n}{\rm
Re}\,z_{i})}\|f^{b}\|_{\infty},\qquad\mathbf{z}\in{\mathbb{C}}_{+}^{n}.$

Indeed, $e_{-\tau}f\in L^{\infty}(\mathbb{R}^{n})$ and ${\rm
sp}\,(e_{-\tau}f)\subseteq\mathbb{R}^{n}_{+}$. If
$g({\bm{\alpha}}+i{\bm{\beta}})=(P_{{\bm{\beta}}}*(e_{-\tau}f)^{b})({\bm{\alpha}}),$
${\bm{\alpha}}+i{\bm{\beta}}\in\mathbb{C}^{n}_{+}$, then by the above $g\in
H^{\infty}(\mathbb{C}_{+}^{n})$ and $(e_{\tau}g)^{b}=f^{b}$. Since both $f$
and $e_{\tau}g$ are in $H^{\infty}(\mathbb{C}^{n}_{+})$ we infer from (8.1)
that $e_{\tau}(\mathbf{z})g(\mathbf{z})=f(\mathbf{z})$ for all
$\mathbf{z}\in\mathbb{C}^{n}_{+}$. Taking into account (8.1), we obtain (8.3)
which is well-known for $n=1$. It follows from a standard Fourier
characterisation of $H^{\infty}(\mathbb{C}_{+})$ and it was used essentially
in [6], but its multivariate counterpart seems not to have been noted in the
literature.

We will use an elementary variant of Bernstein’s inequality, as follows (see
[19, Theorem 7.3.1] or [28, Chapter 3, p.116]). If $f\in
H^{\infty}([0,\sigma]^{n})$ then $f$ is an entire function of exponential type
$(\sigma,\dots,\sigma)$, as defined in [28, p.98] and for all
$\mathbf{z}={\bm{\alpha}}+i{\bm{\beta}}\in\mathbb{C}^{n}_{+}$ and
$\Omega\in\mathcal{P}_{n}$

(8.4)
$\sup_{{\bm{\beta}}\in\mathbb{R}^{n}}|\mathcal{D}_{\Omega}f({\bm{\alpha}}+i{\bm{\beta}})|\leq\sigma^{|\Omega|}\sup_{{\bm{\beta}}\in\mathbb{R}^{n}}\,|f({\bm{\alpha}}+i{\bm{\beta}})|.$

In the following lemmas, we show that functions of various classes belong to
$\mathcal{B}_{0}^{n}$, and we give estimates for their norms. Then Proposition
2.6 shows that $f\in\mathcal{B}^{n}$ and $\|f\|_{\mathcal{B}^{n}}\leq
2^{n}\|f\|_{\mathcal{B}_{0}^{n}}$. An alternative estimate of the
$\mathcal{B}^{n}$-norms can be obtained by an extension of the method used for
the $\mathcal{B}_{0}^{n}$-norms.

### Holomorphic extensions to the left

We will show here that if $f\in\mathcal{B}^{n}$ extends to a bounded analytic
function in a larger half-space, then some damped versions of $f$ are in
$\mathcal{B}_{0}^{n}$ with norm dominated by an $H^{\infty}$-norm of $f$. For
$n=1$, functions of this type were considered in [6, Section 5.2].

Let $\omega>0$ and
$H^{\infty}_{\omega}=H^{\infty}(\\{\mathbf{z}\in\mathbb{C}^{n}:\operatorname{Re}z_{j}>-\omega,\,j\in
I_{n}\\})$. If $f\in H^{\infty}_{\omega}$ and $\Omega\in\mathcal{P}_{n}$, then
Cauchy’s inequality gives

(8.5)
$|\mathcal{D}_{\Omega}f(\mathbf{z})|\leq\frac{\|g\|_{H^{\infty}_{\omega}}}{2^{k}|\omega+{\bm{\alpha}}_{\Omega}|},\qquad\mathbf{z}={\bm{\alpha}}+i{\bm{\beta}}\in\mathbb{C}^{n}_{+},$

where $k=|\Omega|$ and
$|\omega+{\bm{\alpha}}_{\Omega}|=\prod_{j\in\Omega}(\omega+\alpha_{j})$.

######  Lemma 8.1.

Let $f\in H^{\infty}_{\omega}$, $\omega>0$ and let $\nu>0$,
$\lambda\in{\mathbb{C}}_{+}$, $m=\min\\{\omega,\operatorname{Re}\lambda\\}$.
Let

(8.6)
$R^{\nu}_{\lambda}(\mathbf{z}):=\prod_{j=1}^{n}(\lambda+z_{j})^{-\nu},\qquad
S^{\nu}_{\lambda}(\mathbf{z}):=(\lambda+z_{1}+\cdots+z_{n})^{-\nu},\quad\mathbf{z}\in\mathbb{C}_{+}^{n}.$

Then $R^{\nu}_{\lambda}f\in\mathcal{B}^{n}_{0}$ and
$S^{\nu}_{\lambda}f\in\mathcal{B}^{n}_{0}$. Moreover,

$\displaystyle\|R^{\nu}_{\lambda}f\|_{\mathcal{B}_{0}^{n}}\leq\|f\|_{H^{\infty}_{\omega}}\left(\frac{1}{2\nu}+\frac{1}{m^{\nu}}\right)^{n},\quad\|S^{\nu}_{\lambda}f\|_{\mathcal{B}_{0}^{n}}\leq\frac{\|f\|_{H^{\infty}_{\omega}}}{m^{\nu}}\left(\frac{n}{2\nu}+1\right)^{n}.$

###### Proof.

From (2.3) and (8.5), we have

$\displaystyle\|R_{\lambda}^{\nu}f\|_{\mathcal{B}_{0}^{n}}$
$\displaystyle\leq\sum_{\Omega\in\mathcal{P}_{n}}\int_{{\mathbb{R}}_{+}^{n}}\,\sup_{\mathbf{z}\in
W_{{\bm{\alpha}}_{I_{n}}}}\,|(D_{\Omega}R_{\lambda}^{\nu})({\mathbf{z}})||(D_{\Omega^{c}}f)(\mathbf{z})|\,d{\bm{\alpha}}$
$\displaystyle\leq\|f\|_{H^{\infty}_{\omega}}\sum_{\Omega\in\mathcal{P}_{n}}\frac{\nu^{|\Omega|}}{2^{n-|\Omega|}}\int_{{\mathbb{R}}^{|\Omega|}_{+}}\frac{d{\bm{\alpha}}}{|m+{\bm{\alpha}}_{\Omega}|^{1+\nu}}$
$\displaystyle=\|f\|_{H^{\infty}_{\omega}}\sum_{j=0}^{n}\frac{\binom{n}{j}}{(2\nu)^{n-j}}\frac{1}{m^{\nu
j}}=\|f\|_{H^{\infty}_{\omega}}\left(\frac{1}{2\nu}+\frac{1}{m^{\nu}}\right)^{n}.$

By a similar argument, we have (with $d{\bm{t}}_{n-j}=dt_{1}\dots dt_{n-j}$),

$\displaystyle\|S_{\lambda}^{\nu}f\|_{\mathcal{B}_{0}^{n}}$
$\displaystyle\leq\|f\|_{H^{\infty}_{\omega}}\sum_{j=0}^{n}\frac{\binom{n}{j}}{2^{n-j}}\prod_{k=0}^{j-1}(\nu+k)\int_{{\mathbb{R}}_{+}^{n}}\big{(}m+\sum_{k=1}^{n}t_{k}\big{)}^{-(\nu+j)}\prod_{k=1}^{n-j}(m+t_{k})^{-1}\,d{\bm{t}}$
$\displaystyle=\|f\|_{H^{\infty}_{\omega}}\sum_{j=0}^{n}\frac{\binom{n}{j}}{2^{n-j}}\int_{{\mathbb{R}}_{+}^{n-j}}\big{(}m+\sum_{k=1}^{n-j}t_{k}\big{)}^{-\nu}\prod_{k=1}^{n-j}(m+t_{k})^{-1}\,d{\bm{t}}_{n-j}.$

By replacing $t_{k}$ by $mt_{k}$, taking the inequality
$\prod_{k=1}^{n-j}(1+t_{k})\leq\big{(}1+\sum_{k=1}^{n-j}t_{k}\big{)}^{n-j}$
and raising it to the power $-\nu/(n-j)$, we arrive at

$\displaystyle\|S_{\lambda}^{\nu}f\|_{\mathcal{B}_{0}^{n}}$
$\displaystyle\leq\frac{\|f\|_{H^{\infty}_{\omega}}}{m^{\nu}}\sum_{j=0}^{n}\frac{\binom{n}{j}}{2^{n-j}}\int_{{\mathbb{R}}_{+}^{n-j}}\prod_{k=1}^{n-j}(1+t_{k})^{-(1+\nu/(n-j))}\,d{\bm{t}}_{n-j}$
$\displaystyle=\frac{\|f\|_{H^{\infty}_{\omega}}}{m^{\nu}}\sum_{j=0}^{n}\binom{n}{j}\left(\frac{n-j}{2\nu}\right)^{n-j}\leq\frac{\|f\|_{H^{\infty}_{\omega}}}{m^{\nu}}\left(\frac{n}{2\nu}+1\right)^{n}.\qed$

Now we show that the damping in Lemma 8.1 is not necessary if ${\rm sp}(f)$ is
separated from zero.

######  Lemma 8.2.

Let $f\in H^{\infty}([\tau,\infty)^{n})\cap H^{\infty}_{\omega}$, $\tau>0$,
$\omega>0$. Then $f\in\mathcal{B}^{n}_{0}$, and

(8.7)
$\|f\|_{\mathcal{B}_{0}^{n}}\leq\|f\|_{H^{\infty}_{\omega}}e^{-n\omega\tau}\left(1+\frac{1}{2}\log\left(1+\frac{1}{\tau\omega}\right)\right)^{n}.$

###### Proof.

By (8.3), there exists $g\in H^{\infty}_{\omega}$ such that
$f(\mathbf{z})=e^{-\tau(z_{1}+\cdots+z_{n})}g(\mathbf{z}),\,\mathbf{z}\in{\mathbb{C}}_{+}^{n},$
and thus
$\|g\|_{H_{\omega}^{\infty}}=e^{-n\omega\tau}\|f\|_{H^{\infty}_{\omega}}$.
From this and (8.5), we infer that

$\displaystyle|\mathcal{D}_{n}f(\mathbf{z})|$
$\displaystyle\leq\sum_{\Omega\in\mathcal{P}_{n}}|(D_{\Omega}e^{-\tau(z_{1}+\cdots+z_{n})})(D_{\Omega^{c}}g(\mathbf{z}))|$
$\displaystyle\leq
e^{-\tau(\alpha_{1}+\cdots+\alpha_{n})}\sum_{\Omega\in\mathcal{P}_{n}}\left(\frac{\tau^{|\Omega|}}{2^{n-|\Omega}|\omega+{{\bm{\alpha}}}_{\Omega^{c}}|}\right)\|g\|_{H^{\infty}_{\omega}},$

where $z_{j}=\alpha_{j}+i\beta_{j}$. Since the variables in the integrals
below over ${\mathbb{R}}_{+}^{n}$ split as the product of $j$ integrals of one
function and $n-j$ integrals of another function, we see that

$\displaystyle\|f\|_{\mathcal{B}_{0}^{n}}$
$\displaystyle\leq\|g\|_{H^{\infty}_{\omega}}\sum_{\Omega\in\mathcal{P}_{n}}\,\frac{\tau^{|\Omega|}}{2^{n-|\Omega|}}\int_{{\mathbb{R}}_{+}^{n}}e^{-\tau\sum_{j=1}^{n}\alpha_{j}}\left(\frac{1}{|\omega+{\bm{\alpha}}_{\Omega^{c}}|}\right)\,d{\bm{\alpha}}$
$\displaystyle=\|g\|_{H^{\infty}_{\omega}}\sum_{j=0}^{n}\binom{n}{j}\frac{1}{2^{n-j}}\left(\int_{0}^{\infty}\tau
e^{-\tau\alpha}\,d\alpha\right)^{j}\left(\int_{0}^{\infty}e^{-\tau\alpha}(\omega+\alpha)^{-1}\,d\alpha\right)^{n-j}.$

The first integral in the line above is equal to $1$, and the second integral
is shown in the proof of [6, Lemma 3.2(2)] to be less than
$\log(1+(\tau\omega)^{-1})$, so

$\displaystyle\|f\|_{\mathcal{B}_{0}^{n}}$
$\displaystyle\leq\|g\|_{H^{\infty}_{\omega}}\sum_{j=0}^{n}\binom{n}{j}\\!\left(\frac{1}{2}\log\left(1+\frac{1}{\tau\omega}\right)\\!\right)^{n-j}\\!=\|g\|_{H^{\infty}_{\omega}}\\!\left(1+\frac{1}{2}\log\left(\\!1+\frac{1}{\tau\omega}\right)\\!\right)^{n}\\!\\!.\qed$

### Functions with restricted spectrum

Instead of assuming that $f\in H^{\infty}_{\omega}$, we now assume that ${\rm
sp}(f)$ is separated from zero and infinity and we obtain an estimate of
$\|f\|_{\mathcal{B}^{n}}$ in terms of $\|f\|_{\infty}$. We need the following
technical lemma. For ${\bm{t}}=(t_{1},\dots,t_{k})\in(0,\infty)^{k}$, let
$S_{k}({\bm{t}})=\sum_{j=1}^{k}t_{j}$ and
$P_{k}({\bm{t}})=\prod_{j=1}^{k}t_{j}$.

######  Lemma 8.3.

For $k\in{\mathbb{N}}$ and $a\in(0,1)$, let

$J_{k}(a):=\int_{{\mathbb{R}}_{+}^{k}}\,\frac{d{\bm{t}}}{P_{k}({\bm{t}})+ae^{S_{k}({\bm{t}})}}.$

Then

(8.8) $J_{k}(a)\leq 2^{2k}\log^{k}\left(1+\frac{1}{a}\right).$

###### Proof.

We argue by induction. If $k=1,$ then (8.8) has been proved in [6, p.37,
(2.12)]. Next, assume that (8.8) holds for some $k\geq 1$. Note that $J_{k}$
is a decreasing function, and by Fubini’s theorem,

$J_{k+1}(a)=\int_{0}^{\infty}\int_{{\mathbb{R}}_{+}^{k}}\frac{d{\bm{t}}\,d\tau}{P_{k}({\bm{t}})\tau+ae^{S_{k}({\bm{t}})}e^{\tau}}=\int_{0}^{\infty}J_{k}(ae^{\tau}/\tau)\,\frac{d\tau}{\tau}.$

Using the monotonicity of $J_{k}$, a change of variable $u=e^{-\tau}$, and the
inequalities $tJ_{k}(t)\leq 1$, and $(4\log 2)^{k+1}\geq 2(k+1)$ if $k\geq 1$,
we infer that

$\displaystyle J_{k+1}(a)$
$\displaystyle\leq\int_{0}^{1}J_{k}(a/\tau)\frac{d\tau}{\tau}+\int_{1}^{\infty}J_{k}(ae^{\tau})\,d\tau=\int_{0}^{1}J_{k}(a/\tau)\frac{d\tau}{\tau}+\int_{0}^{1/e}J_{k}(a/u)\frac{du}{u}$
$\displaystyle\leq
2\int_{0}^{1}J_{k}(1/s)\frac{ds}{s}+2\int_{1}^{1/a}J_{k}(1/s)\frac{ds}{s}\leq
2+2^{2k+1}\int_{1}^{1/a}\log^{k}(1+s)\,\frac{ds}{s}$ $\displaystyle\leq
2+2^{2k+2}\int_{1}^{1/a}\frac{\log^{k}(1+s)}{1+s}\,ds=2+\frac{2^{2k+2}}{k+1}\left(\log^{k+1}(1+1/a)-\log^{k+1}2\right)$
$\displaystyle\leq 2^{2(k+1)}\log^{k+1}(1+1/a),$

so (8.8) is true for $k+1$. The assertion follows. ∎

The following lemma was proved in [6, Lemma 2.5] for $n=1$ and in [21, Lemma
4.1.1] for $n=2$. The functions considered in those cases played crucial roles
in those papers.

######  Lemma 8.4.

Let $f\in H^{\infty}([\epsilon,\sigma]^{n})$, where $0<\epsilon<\sigma$. Then
$f\in\mathcal{B}^{n}_{0}$ and

(8.9) $\|f\|_{\mathcal{B}_{0}^{n}}\leq
2^{n+1}\|f\|_{\infty}\left(\log\left(1+\left(\frac{2\sigma}{\epsilon}\right)^{n}\right)\right)^{n}.$

###### Proof.

From Bernstein’s inequality, we have

(8.10)
$|\mathcal{D}_{n}f({\bm{\alpha}}+i{\bm{\beta}})|\leq\sigma^{n}\sup_{\bm{\beta}\in{\mathbb{R}}^{n}}\,|f({\bm{\alpha}}+i{\bm{\beta}})|\leq\sigma^{n}e^{-\epsilon
S({\bm{\alpha}})}\|f\|_{\infty}.$

From Cauchy’s inequality,

$|\mathcal{D}_{n}f({\bm{\alpha}}+i{\bm{\beta}})|\leq\frac{\|f\|_{\infty}}{2^{n}P_{n}({\bm{\alpha}})}.$

Thus

$\displaystyle|\mathcal{D}_{n}f({\bm{\alpha}}+i{\bm{\beta}})|$
$\displaystyle\leq\|f\|_{\infty}\min\left\\{\sigma^{n}e^{-\epsilon
S_{n}({\bm{\alpha}})},2^{-n}P_{n}({\bm{\alpha}})^{-1}\right\\}\leq\frac{2\|f\|_{\infty}}{2^{n}P_{n}({\bm{\alpha}})+\sigma^{-n}e^{\epsilon
S_{n}({\bm{\alpha}})}}.$

From Lemma, 8.3,

$\displaystyle\int_{{\mathbb{R}}_{+}^{n}}\,\frac{d{\bm{\alpha}}}{P_{n}({\bm{\alpha}})+\sigma^{-n}e^{\epsilon
S_{n}({\bm{\alpha}})}}=J_{n}\left(\left(\frac{\epsilon}{2\sigma}\right)^{n}\right)\leq
2^{n}\left(\log\left(1+\left(\frac{2\sigma}{\epsilon}\right)^{n}\right)\right)^{n}.$

Now (8.9) follows. ∎

Lemmas 8.1, 8.2 and 8.4 imply the following operator norm-estimates for
$\mathcal{B}^{n}$-functions of $n$-tuples of operators satisfying the full
(GSF) condition. They were obtained in [6] for $n=1$, providing direct
alternative proofs for the results in [15], [16], [30] and their improvements.
The estimates quantify a deviation of the $\mathcal{B}^{n}$-calculus from the
bounded $H^{\infty}$-calculus, and they simplify operator norm-estimates in
several situations of interest. In statements (i) and (ii), $f({\mathcal{A}})$
is not necessarily a bounded operator, but it can be defined as a closed
operator in the extended half-plane calculus for several variables. The proofs
are consequences of Lemmas 8.1, 8.2 and 8.4. For (ii), one uses (8.2).

######  Theorem 8.5.

Let $\mathcal{A}$ be an $n$-tuple of commuting operators on a Banach space
$X$, satisfying the full (GSF) condition, and let $\gamma_{\mathcal{A},0}$ be
the norm of $\Phi_{A}$ on $\mathcal{B}_{0}^{n}$. Then the following hold.

  1. (i)

Let $f\in H^{\infty}_{\omega}$, $\omega>0$, and let $\nu>0$,
$\lambda\in{\mathbb{C}}_{+}$, and $m=\min\\{\omega,{\rm Re}\,\lambda\\}$. Let
$R_{\lambda}^{\nu}$ and $S_{\lambda}^{\nu}$ be as in (8.6). Then

$\|(R_{\lambda}^{\nu}f)({\mathcal{A}})\|\leq\gamma_{\mathcal{A},0}\left(\frac{1}{2\nu}+\frac{1}{m}\right)^{n}\|f\|_{H_{\omega}^{\infty}}.$

and

$\|(S_{\lambda}^{\nu}f)({\mathcal{A}})\|\leq\gamma_{\mathcal{A},0}m^{-\nu}\left(\frac{n}{2\nu}+1\right)^{n}\|f\|_{H^{\infty}_{\omega}}.$

  2. (ii)

Let $g\in H^{\infty}_{\omega}$, $\omega>0$ and $\tau>0$. Then

$\|(ge_{-\tau})({\mathcal{A}})\|\leq\gamma_{\mathcal{A},0}e^{-n\omega\tau}\left(1+\frac{1}{2}\log\left(1+\frac{1}{\tau\omega}\right)\right)^{n}\|g\|_{H^{\infty}_{\omega}}.$

  3. (iii)

Let $f\in H^{\infty}([\epsilon,\sigma]^{n})$, $0<\epsilon<\sigma$. Then

(8.11) $\|f(\mathcal{A})\|\leq
2^{n+1}\gamma_{\mathcal{A},0}\left(\log\left(1+\left(\frac{2\sigma}{\epsilon}\right)^{n}\right)\right)^{n}\|f\|_{\infty}.$

##  9\. Appendix

For the sceptical reader, we will justify the application of Fubini’s theorem
in the proof of Lemma 5.1 by showing that

$J_{t}:=\int_{{\mathbb{C}}_{+}^{m}}\int_{{\mathbb{C}}_{+}^{k}}|(\mathcal{D}_{\Omega_{f}}f)({\bm{\lambda}})|\,|(\mathcal{D}_{\Omega
g}g)({\bm{\nu}})|\,S_{t}({\bm{\lambda}},{\bm{\nu}})\,dV_{k}({\bm{\lambda}})dV_{m}({\bm{\nu}})<\infty,$

where $f$ and $g$ are elementary functions in $\mathcal{B}^{n}$ with supports
$\Omega_{f}$ and $\Omega_{g}$ with cardinalities $k$ and $m$ respectively, and
$\Omega_{f}\cup\Omega_{g}=I_{n}$. Here ${\bm{\lambda}}$ and ${\bm{\nu}}$ are
indexed by $\Omega_{f}$ and $\Omega_{g}$, respectively, and

$S_{t}({\bm{\lambda}},{\bm{\nu}})=\int_{{\mathbb{C}}_{+}^{n}}|\langle\mathcal{K}_{n}(\mathcal{A},\overline{\mathbf{z}})x,x^{*}\rangle|\left|\mathcal{D}_{n}\left(\mathcal{K}_{k}(\mathbf{z}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{{\bm{\lambda}}})\mathcal{K}_{m}(\mathbf{z}_{\Omega_{g}}+{\bm{t}}_{\Omega_{g}},\overline{{\bm{\nu}}})\right)\right|dV_{n}(\mathbf{z})$

where $\mathcal{D}_{n}$ denotes differentiation with respect to all the
$z_{j}$ variables (once each).

We may assume that $\|x\|=\|x^{*}\|=1$ and put
$\beta_{j}=\operatorname{Re}\lambda_{j}$, $\eta_{l}=\operatorname{Re}\nu_{l}$.
Using the (GSF) condition for ${\mathcal{A}}$, we can estimate:

$\displaystyle S_{t}({\bm{\lambda}},{\bm{\nu}})$
$\displaystyle\leq\gamma_{\mathcal{A}}\int_{{\mathbb{R}}_{+}^{n}}\sup_{\operatorname{Re}z_{j}=\alpha_{j}}\left|\mathcal{D}_{n}(\mathcal{K}_{k}(\mathbf{z}_{\Omega_{f}}+{\bm{t}}_{\Omega_{f}},\overline{\bm{\lambda}})\mathcal{K}_{m}(\mathbf{z}_{\Omega_{g}}+{\bm{t}}_{\Omega_{g}},\overline{{\bm{\nu}}}))\right|\,d{\bm{\alpha}}$
$\displaystyle\leq\gamma_{\mathcal{A}}\bigg{|}\int_{{\mathbb{R}}_{+}^{n}}\mathcal{D}_{n}\left(\mathcal{K}_{k}({\bm{\alpha}}_{\Omega_{f}}+\bm{t}_{\Omega_{f}},{\bm{\beta}}_{\Omega_{f}})\mathcal{K}_{m}({\bm{\alpha}}_{\Omega_{g}}+\bm{t}_{\Omega_{g}},\bm{\eta}_{\Omega_{g}})\right)\,d{\bm{\alpha}}\bigg{|}$
$\displaystyle=\gamma_{\mathcal{A}}\big{|}\mathcal{K}_{k}(\bm{t}_{\Omega_{f}},{\bm{\beta}}_{\Omega_{f}})\mathcal{K}_{m}(\bm{t}_{\Omega_{g}},\bm{\eta}_{\Omega_{g}})\big{|}\leq\gamma_{\mathcal{A}}\left(\frac{2}{\pi}\right)^{k+m}\prod_{j\in\Omega_{f}}\frac{1}{2t\beta_{j}}\prod_{l\in\Omega_{g}}\frac{1}{2t\eta_{l}}.$

The inequality in the second line is obtained by straightforward estimates on
the integrand on the function. Moving the absolute value signs outside the
integral is justified, because the integrand is real-valued, and its sign
depends only on $k$ and $m$. The equality in the third line comes from the
fundamental theorem of calculus for each of the variables. The final
inequality is straightforward. Now

$\displaystyle J_{t}$ $\displaystyle\leq\frac{\gamma_{\mathcal{A}}}{(\pi
t)^{k+m}}\int_{{\mathbb{R}}_{+}^{m}}\int_{{\mathbb{R}}_{+}^{k}}\sup_{\operatorname{Re}\lambda_{j}=\beta_{j},\,\operatorname{Re}\nu_{l}=\eta_{l}}\,|(\mathcal{D}_{\Omega_{f}}f)({\bm{\lambda}})||(\mathcal{D}_{\Omega_{g}}g)({\bm{\nu}})|\,d{\bm{\beta}}d\bm{\eta}$
$\displaystyle=\frac{\gamma_{\mathcal{A}}}{(\pi
t)^{k+m}}\|f\|_{\mathcal{B}_{\Omega_{f},0}}\,\|g\|_{\mathcal{B}_{\Omega_{g},0}}.$

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