# Lipschitz interpolative nonlinear ideal procedure

M. A. S. SALEH
Department of Mathematics and Computer Applications,
College of Science, Al-Nahrain University, Baghdad, Iraq.
E-mail<EMAIL_ADDRESS>
###### Abstract

We treat the general theory of nonlinear ideals and extend as many notions as
possible from the linear theory to the nonlinear theory. We define nonlinear
ideals with special properties which associate new non-linear ideals to given
ones and establish several properties and characterizations of them. Building
upon the results of U. Matter we define a Lipschitz interpolative nonlinear
ideal procedure between metric spaces and Banach spaces and establish this
class of Lipschitz operators is an injective Banach nonlinear ideal and show
several standard basic properties for such class. Extending the work of J. A.
López Molina and E. A. Sánchez Pérez we define a Lipschitz
$\left(p,\theta,q,\nu\right)$-dominated operators for $1\leq p,q<\infty$;
$0\leq\theta,\nu<1$ and establish several characterizations. Afterwards we
generalize a notion of Lipschitz interpolative nonlinear ideal procedure
between arbitrary metric spaces and prove its a nonlinear ideal. Finally, we
present certain basic counter examples of Lipschitz interpolative nonlinear
ideal procedure between arbitrary metric spaces.

2010 AMS Subject Classification. Primary 47L20; Secondary 26A16, 47A57.

##  1 Notations and Preliminaries

We introduce concepts and notations that will be used in this article. The
letters $E$, $F$ and $G$ will denote Banach spaces. The closed unit ball of a
Banach space $E$ is denoted by $B_{E}$. The dual space of $E$ is denoted by
$E^{*}$. The class of all bounded linear operators between arbitrary Banach
spaces will be denoted by $\mathfrak{L}$. The symbols $\mathbb{K}$ and
$\mathbb{N}$ stand for the set of all scalar field and the set of all natural
numbers, respectively. The symbols $W(B_{E^{*}})$ and $W(B_{X^{\\#}})$ stand
for the set of all Borel probability measures defined on $B_{E^{*}}$ and
$B_{X^{\\#}}$, respectively. The value of $a$ at the element $x$ is denoted by
$\left\langle x,a\right\rangle$. We put
$E^{\text{inj}}:=\ell_{\infty}(B_{E^{*}})$ and $J_{E}x:=\left(\left\langle
x,a\right\rangle\right)$ for $x\in E$. Clearly $J_{E}$ is a metric injection
from $E$ into $E^{\text{inj}}$. Let $0<p<\infty$. The Banach space of all
absolutely $p$-summable sequences $\mathbf{x}=(x_{j})_{j\in\mathbb{N}}$, where
$x_{j}\in E$, is denoted by $\ell_{p}(E)$. We put

$\left\|\mathbf{x}\Big{|}\ell_{p}(E)\right\|=\Bigg{[}\sum\limits_{j=1}^{\infty}\left\|x_{j}\right\|^{p}\Bigg{]}^{\frac{1}{p}}<\infty.$

The Banach space of all weakly absolutely $p$-summable sequences
$\mathbf{x}\subset E$, is denoted by $\ell_{p}^{w}(E)$. We put

$\left\|\mathbf{x}\Big{|}\ell_{p}^{w}(E)\right\|=\sup\limits_{a\in
B_{E^{*}}}\Bigg{[}\sum\limits_{j=1}^{\infty}\left|\left\langle
x_{j},a\right\rangle\right|^{p}\Bigg{]}^{\frac{1}{p}}.$ (1)

For the triple sequence
$(\sigma,x^{\prime},x^{\prime\prime})\subset\mathbb{R}\times X\times X$. We
put

$\left\|(\sigma,x^{\prime},x^{\prime\prime})\Big{|}\ell_{p}(\mathbb{R}\times
X\times
X)\right\|=\Bigg{[}\sum\limits_{j=1}^{\rm\infty}\left|\sigma_{j}\right|^{p}d_{X}(x^{\prime}_{j},x^{\prime\prime}_{j})^{p}\Bigg{]}^{\frac{1}{p}}.$

And

$\left\|(\sigma,x^{\prime},x^{\prime\prime})\Big{|}\ell_{p}^{L,w}(\mathbb{R}\times
X\times X)\right\|=\sup\limits_{f\in
B_{{X}^{\\#}}}\Bigg{[}\sum\limits_{j=1}^{\infty}\left|\sigma_{j}\right|^{p}\left|\left\langle
f,x^{\prime}_{j}\right\rangle-\left\langle
f,x^{\prime\prime}_{j}\right\rangle\right|^{p}\Bigg{]}^{\frac{1}{p}}.$

For $0\leq\theta<1$ and $1\leq p<\infty$ we define

$\left\|\mathbf{x}\Big{|}\delta_{p,\theta}(E)\right\|=\underset{\xi\in
B_{E^{\ast}}}{\sup}\left(\underset{j=1}{\overset{\infty}{\sum}}\left(\left|\left\langle
x_{j},\xi\right\rangle\right|^{1-\theta}\left\|x_{j}\right\|^{\theta}\right)^{\frac{p}{1-\theta}}\right)^{\frac{1-\theta}{p}}.$

Also for all sequences
$(\sigma,x^{\prime},x^{\prime\prime})\subset\mathbb{R}\times X\times X$, we
define

$\left\|(\sigma,x^{\prime},x^{\prime\prime})\Big{|}\delta_{p,\theta}^{L}(\mathbb{R}\times
X\times X)\right\|=\sup\limits_{f\in
B_{X^{\\#}}}\left[\sum\limits_{j=1}^{\infty}\left(\left|\sigma_{j}\right|\left|f(x_{j}^{\prime})-f(x_{j}^{\prime\prime})\right|^{1-\theta}d_{X}(x_{j}^{\prime},x_{j}^{\prime\prime})^{\theta}\right)^{\frac{p}{1-\theta}}\right]^{\frac{1-\theta}{p}}$

Recall that the definition of an operator ideal between arbitrary Banach
spaces of A. Pietsch [15] and [13] is as follows. Suppose that, for every pair
of Banach spaces $E$ and $F$, we are given a subset $\mathfrak{A}(E,F)$ of
$\mathfrak{L}(E,F)$. The class

$\mathfrak{A}:=\bigcup_{E,F}\mathfrak{A}(E,F)$

is said to be an operator ideal, or just an ideal, if the following conditions
are satisfied:

  1. $\bf(OI_{0})$

$a^{*}\otimes e\in\mathfrak{A}(E,F)$ for $a^{*}\in E^{*}$ and $e\in F$.

  2. $\bf(OI_{1})$

$S+T\in\mathfrak{A}(E,F)$ for $S,\>T\in\mathfrak{A}(E,F)$.

  3. $\bf(OI_{2})$

$BTA\in\mathfrak{A}(E_{0},F_{0})$ for $A\in\mathfrak{L}(E_{0},E)$,
$T\in\mathfrak{A}(E,F)$, and $B\in\mathfrak{L}(F,F_{0})$.

Condition $\bf(OI_{0})$ implies that $\mathfrak{A}$ contains nonzero
operators.

######  Remark 1.

The normed (Banach) operator ideal is designated by
$\left[\textfrak{A},\mathbf{A}\right]$.

##  2 Introduction

One important example of operator ideals is the class of $p$-summing operators
defined by A. Pietsch [14] as follow: A bounded operator $T$ from $E$ into $F$
is called $p$-summing if and only if there is a constant $C\geq 0$ such that

$\left\|(Tx_{j})_{j=1}^{m}\Big{|}\ell_{p}(F)\right\|\leq
C\cdot\left\|\left(x_{j}\right)_{j=1}^{m}\Big{|}\ell_{p}^{w}(E)\right\|$ (2)

for arbitrary sequence $\left(x_{j}\right)_{j=1}^{m}$ in $E$ and
$m\in\mathbb{N}$. Let us denote by $\Pi_{p}(E,F)$ the class of all $p$-summing
operators from $E$ into $F$ with $\pi_{p}(T)$ summing norm of $T$ is the
infimum of such constants $C$.

J. D. Farmer and W. B. Johnson [8] defined a true extension of the linear
concept of $p$-summing operators as follows: a Lipschitz operator
$T\in\operatorname{Lip}(X,Y)$ is called Lipschitz $p$-summing map if there is
a nonnegative constants $C$ such that for all $m\in\mathbb{N}$, any sequences
$x^{\prime}$, $x^{\prime\prime}$ in $X$ and $\lambda$ in $\mathbb{R}^{+}$, the
inequality

$\left\|(\lambda,Tx^{\prime},Tx^{\prime\prime})\Big{|}\ell_{p}(\mathbb{R}\times
X\times X)\right\|\leq
C\cdot\left\|(\lambda,x^{\prime},x^{\prime\prime})\Big{|}\ell_{p}^{L,w}(\mathbb{R}\times
X\times X)\right\|$

holds. Let us denote by $\Pi_{p}^{L}(X,Y)$ the class of all Lipschitz
$p$-summing maps from $X$ into $Y$ with $\pi_{p}^{L}(T)$ Lipschitz summing
norm of $T$ is the infimum of such constants $C$.

Jarchow and Matter [10] defined a general interpolation procedure for creating
a new operator ideal between arbitrary Banach spaces. Also U. Matter defined
in his seminal paper [12] a new class of interpolative ideal procedure as
follows: let $0\leq\theta<1$ and $\left[\textfrak{A},\mathbf{A}\right]$ be a
normed operator ideal. A bounded operator $T$ from $E$ into $F$ belongs to
$\textfrak{A}_{\theta}(E,F)$ if there exist a Banach space $G$ and a bounded
operator $S\in\textfrak{A}(E,G)$ such that

$\left\|Tx|F\right\|\leq\left\|Sx|G\right\|^{1-\theta}\cdot\left\|x\right\|^{\theta},\
\ \forall\>x\in E.$ (3)

For each $T\in\textfrak{A}_{\theta}(E,F)$, we set

$\mathbf{A}_{\theta}(T):=\inf\mathbf{A}(S)^{1-\theta}$ (4)

where the infimum is taken over all bounded operators $S$ admitted in (3).

######  Proposition 1.

[12] $\left[\textfrak{A}_{\theta},\mathbf{A}_{\theta}\right]$ is an injective
complete quasinormed operator ideal.

U. Matter [12] applied Inequality (3) to the ideal
$\left[\Pi_{p},\pi_{p}\right]$ of absolutely $p$-summing operators and
obtained the injective operator ideal $\left(\Pi_{p}\right)_{\theta}$ which is
complete with respect to the ideal norm $\left(\pi_{p}\right)_{\theta}$ and
established the fundamental theorem of $(p,\theta)$-summing operators for
$1\leq p<\infty$ and $0\leq\theta<1$ as follows:

######  Theorem 1.

[12] Let $T$ be a bounded operator from $E$ into $F$ and $C\geq 0$. The
following are equivalent:

  1. 1.

$T\in\left(\Pi_{p}\right)_{\theta}(E,F)$.

  2. 2.

There exist a constant $C$ and a probability measure $\mu$ on $B_{E^{*}}$ such
that

$\left\|Tx|F\right\|\leq
C\cdot\left(\int\limits_{B_{E^{*}}}\left(\left|\left\langle
x,x^{*}\right\rangle\right|^{1-\theta}\left\|x\right\|^{\theta}\right)^{\frac{p}{1-\theta}}d\mu(x^{*})\right)^{\frac{1-\theta}{p}},\forall\;x\in
E.$

  3. 3.

There exists a constant $C\geq 0$ such that for any
$\left(x_{j}\right)_{j=1}^{m}\subset E,$ and $m\in\mathbb{N}$ we have

$\left\|\left(Tx_{j}\right)_{j=1}^{m}\Big{|}\ell_{\frac{p}{1-\theta}}(F)\right\|\leq
C\cdot\left\|\left(x_{j}\right)_{j=1}^{m}|\delta_{p\theta}(E)\right\|.$

In addition, $\left(\pi_{p}\right)_{\theta}(T)$ is the smallest number $C$ for
which, respectively, (2) and (3) hold.

Another example of operator ideals is the class of $(r,p,q)$-summing operators
defined by A. Pietsch [14, Sec. 17.1.1] as follows: Let $0<r,p,q\leq\infty$
and $\frac{1}{r}\leq\frac{1}{p}+\frac{1}{q}$. A bounded operator $T$ from $E$
into $F$ is called $(r,p,q)$-summing if there is a constant $C\geq 0$ such
that

$\left\|\left(\left\langle
Tx_{j},b_{j}\right\rangle\right)_{j=1}^{m}\Big{|}\ell_{r}\right\|\leq
C\cdot\left\|\left(x_{j}\right)_{j=1}^{m}\Big{|}\ell_{p}^{w}(E)\right\|\left\|\left(b_{j}\right)_{j=1}^{m}\Big{|}\ell_{q}^{w}(F^{*})\right\|$
(5)

for arbitrary sequence $\left(x_{j}\right)_{j=1}^{m}$ in $E$,
$\left(b_{j}\right)_{j=1}^{m}$ in $F^{*}$ and $m\in\mathbb{N}$. Let us denote
by $\mathfrak{P}_{(r,p,q)}(E,F)$ the class of all $(r,p,q)$-summing operators
from $E$ into $F$ with $\textbf{P}_{(r,p,q)}(T)$ summing norm of $T$ is the
infimum of such constants $C$.

######  Proposition 2.

[14, Sec. 17.1.2] $\left[\mathfrak{P}_{(r,p,q)},\textbf{P}_{(r,p,q)}\right]$
is a normed operator ideal.

Let $0<p,q\leq\infty$. A. Pietsch [14] is also defined $(p,q)$-dominated
operator as follows: A bounded operator $T$ from $E$ into $F$ is called
$(p,q)$-dominated if it belongs to the quasi-normed ideal

$\left[\mathcal{D}_{\left(p,q\right)},D_{\left(p,q\right)}\right]:=\left[\mathfrak{P}_{(r,p,q)},\textbf{P}_{(r,p,q)}\right],$

where $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. For a special case, if
$q=\infty$, then
$\left[\mathcal{D}_{\left(p,\infty\right)},D_{\left(p,\infty\right)}\right]:=\left[\Pi_{p},\pi_{p}\right]$.

J.A. López Molina and E. A. Sánchez Pérez [11] established the important
characteristic of $(p,q)$-dominated operator as follows.

######  Proposition 3.

[11] Let $E$ and $F$ be Banach spaces and $T\in\mathfrak{L}(E,F)$. The
following are equivalent:

  1. 1.

$T\in\mathcal{D}_{\left(p,q\right)}(E,F)$.

  2. 2.

There exist a Banach spaces $G$ and $H$, bounded operators
$S_{1}\in\Pi_{p}(E,G)$ and $S_{2}\in\Pi_{q}(F^{*},H)$ and $C>0$ such that

$\left|\left\langle Tx,b\right\rangle\right|\leq
C\left\|S_{1}x\right\|\left\|S_{2}x\right\|,\ \ \forall x\in E,\forall\;b\in
F^{*}.$ (6)

A general example of $(p,q)$-dominated operators is also defined by J.A. López
Molina and E. A. Sánchez Pérez [11] as follows: Let $1\leq p,q<\infty$ and
$0\leq\theta,\nu<1$ such that
$\frac{1}{r}+\frac{1-\theta}{p}+\frac{1-\nu}{q}=1$ with $1\leq r<\infty$. A
bounded operator $T$ from $E$ to $F$ is called
$\left(p,\theta,q,\nu\right)$-dominated if there exist a Banach spaces $G$ and
$H$, a bounded operator $S\in\Pi_{p}(E,G)$, a bounded operator
$R\in\Pi_{q}(F^{*},H)$ and a positive constant $C$ such that

$\left|\left\langle Tx,b^{*}\right\rangle\right|\leq
C\cdot\left\|x\right\|^{\theta}\left\|Sx|G\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|R(b^{*})|H\right\|^{1-\nu}$
(7)

for arbitrary finite sequences x in $X$ and $b^{*}\subset F^{*}$.

Let us denote by $\mathcal{D}_{\left(p,\theta,q,\nu\right)}(E,F)$ the class of
all $\left(p,\theta,q,\nu\right)$-dominated operators from $E$ to $F$ with

$D_{\left(p,\theta,q,\nu\right)}(T)=\inf\left\\{C\cdot\pi_{p}(S)^{1-\theta}\cdot\pi_{q}(R)^{1-\nu}\right\\},$

where the infimum is taken over all bounded operators $S$ and $R$ and constant
$C$ admitted in (7). They also established an important characteristic of
$\left(p,\theta,q,\nu\right)$-dominated operator as follows.

######  Theorem 2.

Let $E$ and $F$ be Banach spaces and $T\in\mathfrak{L}(E,F)$. The following
are equivalent:

  1. $\bf(1)$

$T\in\mathcal{D}_{\left(p,\theta,q,\nu\right)}(E,F)$.

  2. $\bf(2)$

There is a constant $C\geq 0$ and regular probabilities $\mu$ and $\tau$ on
$B_{E^{*}}$ and $B_{F^{**}}$, respectively such that for every $x$ in $X$ and
$b^{*}$ in $F^{*}$ the following inequality holds

$\left|\left\langle Tx,b^{*}\right\rangle\right|\leq
C\cdot\left[\int\limits_{B_{E^{*}}}\left(\left|\left\langle
x,a\right\rangle\right|^{1-\theta}\left\|x\right\|^{\theta}\right)^{\frac{p}{1-\theta}}d\mu(a)\right]^{\frac{1-\theta}{p}}\cdot\left[\int\limits_{B_{F^{**}}}\left(\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{1-\nu}\left\|b^{*}\right\|^{\nu}\right)^{\frac{q}{1-\nu}}d\tau(b^{**})\right]^{\frac{1-\nu}{q}}.$

  3. $\bf(3)$

There exists a constant $C\geq 0$ such that for every finite sequences x in
$X$ and $b^{*}\subset F^{*}$ the inequality

$\left\|\left\langle Tx,b^{*}\right\rangle|\ell_{r^{\prime}}\right\|\leq
C\cdot\left\|\textbf{x}\Big{|}\delta_{p,\theta}(E)\right\|\left\|b^{*}\Big{|}\delta_{q,\nu}(F^{*})\right\|$
(8)

holds.

  4. $\bf(4)$

There are a Banach space $G$, a bounded operator
$A\in\left(\Pi_{p}\right)_{\theta}(X,G)$ and a bounded operator
$B\in\mathfrak{L}(E,F)$ such that
$B^{*}\in\left(\Pi_{q}\right)_{\nu}^{dual}(F^{*},G^{*})$ and $T=BA$.

In this case, $D_{\left(p,\theta,q,\nu\right)}$ is equal to the infimum of
such constants $C$ in either $\bf(2)$, or $\bf(3)$.

We now describe the contents of this paper. In Section 1, we introduce
notations and preliminaries that will be used in this article. In Section 2,
we first present preliminaries of special cases of those operators that map
weakly (Lipschitz) $p$-summable sequences in arbitrary Banach (metric) space
into strongly (Lipschitz) $p$-summable ones in Banach (metric) space these
operators are called (Lipschitz) $p$-summing operators defined by A. Pietsch
[14], J. D. Farmer and W. B. Johnson [8], respectively. Jarchow and Matter
[10] defined a general interpolation procedure to create a new ideal from
given ideals and U. Matter defined a new class of interpolative ideal
procedure in his seminal paper [12]. He established the fundamental
characterize result of $(p,\theta)$-summing operators for $1\leq p<\infty$ and
$0\leq\theta<1$. For $0<p,q\leq\infty$. A. Pietsch [14] defined
$(p,q)$-dominated operator between arbitrary Banach spaces. J. A. López Molina
and E. A. Sánchez Pérez [11] established the fundamental characterize of
$(p,q)$-dominated operator. Afterwards a general example of $(p,q)$-dominated
operators is also defined by J.A. López Molina and E. A. Sánchez Pérez [11].
This class of operators is called $\left(p,\theta,q,\nu\right)$-dominated for
$1\leq p,q<\infty$ and $0\leq\theta,\nu<1$ such that
$\frac{1}{r}+\frac{1-\theta}{p}+\frac{1-\nu}{q}=1$ with $1\leq r<\infty$. They
proved an important characterize of $\left(p,\theta,q,\nu\right)$-dominated
operator. In Section 3, we treat the general theory of nonlinear operator
ideals. The basic idea here is to extend as many notions as possible from the
linear theory to the nonlinear theory. Therefore, we start by recalling the
fundamental concepts of an operator ideal defined by A. Pietsch [14], see also
[15]. Then, we introduce the corresponding definitions for nonlinear operator
ideals in the version close to that of A. Jiménez-Vargas, J. M. Sepulcre, and
Moisés Villegas-Vallecillos [5]. Afterwards, we define nonlinear ideals with
special properties which associate new non-linear ideals to given ones. Again,
this is parallel to the linear theory. For $0<p\leq 1$ we also define a
Lipschitz $p$-norm on nonlinear ideal and prove that the injective hull
$\textfrak{A}^{L}_{\text{inj}}$ is a $p$-normed nonlinear ideal. We generalize
U. Matter’s interpolative ideal procedure for its nonlinear (Lipschitz)
version between metric spaces and Banach spaces and establish these class of
operators is an injective Banach nonlinear ideal as well as we show several
basic properties for such class. Extending the work of J. A. López Molina and
E. A. Sánchez Pérez we define a Lipschitz
$\left(p,\theta,q,\nu\right)$-dominated operators for $1\leq p,q<\infty$ and
$0\leq\theta,\nu<1$ such that
$\frac{1}{r}+\frac{1-\theta}{p}+\frac{1-\nu}{q}=1$ with $1\leq r<\infty$ and
establish several characterizations analogous to linear case of [11] and prove
that the class of Lipschitz $\left(p,\theta,q,\nu\right)$-dominated operators
is a Banach nonlinear ideal under the Lipschitz
$\left(p,\theta,q,\nu\right)$-norm. In Section 4, we define nonlinear operator
ideal concept between arbitrary metric spaces. It is also in the version close
to that defined in [5]. We generalize a notion of Lipschitz interpolative
nonlinear ideal procedure between arbitrary metric spaces and prove its a
nonlinear ideal. Finally, we present certain basic counter examples of
Lipschitz interpolative nonlinear ideal procedure between arbitrary metric
spaces.

##  3 Nonlinear ideals between arbitrary metric spaces and Banach spaces

######  Definition 1.

Suppose that, for every pair of metric spaces $X$ and Banach spaces $F$, we
are given a subset $\textfrak{A}^{L}(X,F)$ of $\operatorname{Lip}(X,F)$. The
class

$\textfrak{A}^{L}:=\bigcup_{X,F}\textfrak{A}^{L}(X,F)$

is said to be a complete $p$-normed (Banach) nonlinear ideal $\left(0<p\leq
1\right)$, if the following conditions are satisfied:

  1. $\bf(\widetilde{PNOI_{0}})$

$g\boxdot e\in\textfrak{A}^{L}(X,F)$ and $\mathbf{A}^{L}\left(g\boxdot
e\right)=\operatorname{Lip}(g)\cdot\left\|e\right\|$ for $g\in X^{\\#}$ and
$e\in F$.

  2. $\bf(\widetilde{PNOI_{1}})$

$S+T\in\textfrak{A}^{L}(X,F)$ and the $p$-triangle inequality holds:

$\mathbf{A}^{L}\left(S+T\right)^{p}\leq\mathbf{A}^{L}(S)^{p}+\mathbf{A}^{L}(T)^{p}\
\text{for}\ S,\>T\in\textfrak{A}^{L}(X,F).$

  3. $\bf(\widetilde{PNOI_{2}})$

$BTA\in\textfrak{A}^{L}(X_{0},F_{0})$ and
$\mathbf{A}^{L}\left(BTA\right)\leq\left\|B\right\|\mathbf{A}^{L}(T)\>\operatorname{Lip}(A)$
for $A\in\operatorname{Lip}(X_{0},X)$, $T\in\textfrak{A}^{L}(X,F)$, and
$B\in\mathfrak{L}(F,F_{0})$.

  4. $\bf(\widetilde{PNOI_{3}})$

All linear spaces $\textfrak{A}^{L}(X,F)$ are complete, where $\mathbf{A}^{L}$
is called a Lipschitz $p$-norm from $\textfrak{A}^{L}$ to $\mathbb{R}^{+}$.

######  Remark 2.

$\bf(1)$

If $p=1$, then $\mathbf{A}^{L}$ is simply called a Lipschitz norm and
$\left[\textfrak{A}^{L},\mathbf{A}^{L}\right]$ is said to be a Banach
nonlinear ideal.

$\bf(2)$

If $\left[\textfrak{A}^{L},\mathbf{A}^{L}\right]$ be a normed nonlinear ideal,
then $\textfrak{A}^{L}\left(X,\mathbb{R}\right)=X^{\\#}$ with
$\operatorname{Lip}(g)=\mathbf{A}^{L}(g),\forall\ \ g\in X^{\\#}$.

######  Proposition 4.

Let $\textfrak{A}^{L}$ be a nonlinear ideal. Then all components
$\textfrak{A}^{L}(X,F)$ are linear spaces.

###### Proof.

By the condition of $\bf(\widetilde{PNOI_{1}})$ it remains to show that
$T\in\textfrak{A}^{L}(X,F)$ and $\lambda\in\mathbb{K}$ imply $\lambda\cdot
T\in\textfrak{A}^{L}(X,F)$. This follows from $\lambda\cdot
T=\left(\lambda\cdot I_{F}\right)\circ T\circ I_{X}$ and
$\bf(\widetilde{PNOI_{2}})$.
∎

######  Proposition 5.

If $\left[\textfrak{A}^{L},\mathbf{A}^{L}\right]$ be a normed nonlinear ideal,
then $\operatorname{Lip}(T)\leq\mathbf{A}^{L}(T)$ for all
$T\in\textfrak{A}^{L}$.

###### Proof.

Let $T$ be an arbitrary Lipschitz operator in $\textfrak{A}^{L}(X,F)$.

$\displaystyle\operatorname{Lip}(T)=\left\|T^{\\#}_{|_{{F}^{*}}}\right\|$
$\displaystyle=\sup\left\\{\operatorname{Lip}(T^{\\#}b^{*}):b^{*}\in
B_{{F}^{*}}\right\\}$
$\displaystyle=\sup\left\\{\operatorname{Lip}\left(b^{*}\circ
T\right):b^{*}\in B_{{F}^{*}}\right\\}$

Now from Remark 2 we have $\operatorname{Lip}(b^{*}\circ
T)=\mathbf{A}^{L}(b^{*}\circ T)$ for $b^{*}\in{F}^{*}$. It follows

$\operatorname{Lip}(T)=\sup\left\\{\mathbf{A}^{L}(b^{*}\circ T):b^{*}\in
B_{{F}^{*}}\right\\}\leq\mathbf{A}^{L}(T).$

∎

###  3.1 Nonlinear Ideals with Special Properties

####  3.1.1 Lipschitz Procedures

A rule

$\text{new}:\mathfrak{A}\longrightarrow\mathfrak{A}^{L}_{new}$

which defines a new nonlinear ideal $\mathfrak{A}^{L}_{new}$ for every ideal
$\mathfrak{A}$ is called a Lipschitz semi-procedure. A rule

$\text{new}:\textfrak{A}^{L}\longrightarrow\textfrak{A}^{L}_{new}$

which defines a new nonlinear ideal $\textfrak{A}^{L}_{new}$ for every
nonlinear ideal $\textfrak{A}^{L}$ is called a Lipschitz procedure.

######  Remark 3.

We now define the following special properties:

  1. $\bf(M^{\prime})$

If $\textfrak{A}^{L}\subseteq\textfrak{B}^{L}$, then
$\textfrak{A}^{L}_{new}\subseteq\textfrak{B}^{L}_{new}$ (strong monotony).

  2. $\bf(M^{\prime\prime})$

If $\mathfrak{A}\subseteq\mathfrak{B}$, then
$\mathfrak{A}^{L}_{new}\subseteq\mathfrak{B}^{L}_{new}$ (monotony).

  3. $\bf(I)$

$\left(\textfrak{A}^{L}_{new}\right)_{new}=\textfrak{A}^{L}_{new}$ for all
$\textfrak{A}^{L}$ (idempotence).

A strong monotone and idempotent Lipschitz procedure is called a Lipschitz
hull procedure if $\textfrak{A}^{L}\subseteq\textfrak{A}^{L}_{new}$ for all
nonlinear ideals.

####  3.1.2 Closed Nonlinear Ideals

Let $\textfrak{A}^{L}$ be a nonlinear ideal. A Lipschitz operator
$T\in\operatorname{Lip}(X,F)$ belongs to the closure $\textfrak{A}^{L}_{clos}$
if there are $T_{1},T_{2},T_{3},\cdots\in\textfrak{A}^{L}(X,F)$ with
$\lim\limits_{n}\operatorname{Lip}\left(T-T_{n}\right)=0$. It is not difficult
to prove the following result.

######  Proposition 6.

$\textfrak{A}^{L}_{clos}$ is a nonlinear ideal.

The following statement is evident.

######  Proposition 7.

The rule

$\text{clos}:\textfrak{A}^{L}\longrightarrow\textfrak{A}^{L}_{clos}$

is a hull Lipschitz procedure.

######  Definition 2.

The nonlinear ideal $\textfrak{A}^{L}$ is called closed if
$\textfrak{A}^{L}=\textfrak{A}^{L}_{clos}$.

######  Proposition 8.

Let $\textfrak{G}^{L}$ be a Lipschitz approximable nonlinear ideal. Then
$\textfrak{G}^{L}$ is the smallest closed nonlinear ideal.

###### Proof.

By the definition of Lipschitz approximable operators in [9] we have
$\textfrak{G}^{L}=\textfrak{F}^{L}_{clos}$. Hence $\textfrak{G}^{L}$ is
closed. Let $\textfrak{A}^{L}$ be a closed nonlinear ideal. Since
$\textfrak{F}^{L}$ is the smallest nonlinear ideal, we obtain from the
monotonicity of the closure procedure

$\textfrak{G}^{L}=\textfrak{F}^{L}_{clos}\subseteq\textfrak{A}^{L}_{clos}=\textfrak{A}^{L}.$

∎

####  3.1.3 Dual Nonlinear Ideals

Let $\mathfrak{A}$ be an ideal. A Lipschitz operator
$T\in\operatorname{Lip}(X,F)$ belongs to the Lipschitz dual ideal
$\mathfrak{A}^{L}_{dual}$ if
$T^{\\#}_{|_{{F}^{*}}}\in\mathfrak{A}(F^{*},X^{\\#})$.

######  Lemma 1.

Let $T$ in $\textfrak{F}^{L}(X,F)$ with $T=\sum\limits_{j=1}^{m}g_{j}\boxdot
e_{j}$. Then $T^{\\#}_{|_{{F}^{*}}}=\sum\limits_{j=1}^{m}\hat{e}_{j}\otimes
g_{j}$, where $e\longmapsto\hat{e}$ is the natural embedding of the space $F$
into its second dual $F^{**}$.

###### Proof.

We have $Tx=\sum\limits_{j=1}^{m}g_{j}(x)e_{j}$ for $x\in X$. So for $b^{*}\in
F^{*}$,

$\left\langle
T^{\\#}_{|_{{F}^{*}}}b^{*},x\right\rangle_{(X^{\\#},X)}=\left\langle
b^{*},Tx\right\rangle_{(F^{*},F)}=\sum\limits_{j=1}^{m}g_{j}(x)b^{*}(e_{j}).$

Hence $T^{\\#}_{|_{{F}^{*}}}b^{*}=\sum\limits_{j=1}^{m}b^{*}(e_{j})g_{j}$.
This proves the statement for $T^{\\#}_{|_{{F}^{*}}}$. ∎

######  Lemma 2.

Let $T,S\in\operatorname{Lip}(X,F)$, $A\in\operatorname{Lip}(X_{0},X)$, and
$B\in\mathfrak{L}(F,F_{0})$. Then

  1. 1.

$\left(T+S\right)^{\\#}_{|_{{F}^{*}}}=T^{\\#}_{|_{{F}^{*}}}+S^{\\#}_{|_{{F}^{*}}}$.

  2. 2.

$\left(BTA\right)^{\\#}_{|_{F^{*}_{0}}}=A^{\\#}T^{\\#}_{|_{{F}^{*}}}B^{*}$.

###### Proof.

For $b^{*}\in F^{*}$ and $x\in X$, we have

$\displaystyle\left\langle\left(T+S\right)^{\\#}_{|_{{F}^{*}}}b^{*},x\right\rangle_{(X^{\\#},X)}$
$\displaystyle=\left\langle b^{*},(T+S)x\right\rangle_{(F^{*},F)}=\left\langle
b^{*},Tx+Sx\right\rangle_{(F^{*},F)}$ $\displaystyle=\left\langle
b^{*},Tx\right\rangle_{(F^{*},F)}+\left\langle
b^{*},Sx\right\rangle_{(F^{*},F)}$ $\displaystyle=\left\langle
T^{\\#}_{|_{{F}^{*}}}b^{*},x\right\rangle_{(X^{\\#},X)}+\left\langle
S^{\\#}_{|_{{F}^{*}}}b^{*},x\right\rangle_{(X^{\\#},X)}.$

Hence
$\left(T+S\right)^{\\#}_{|_{{F}^{*}}}=T^{\\#}_{|_{{F}^{*}}}+S^{\\#}_{|_{{F}^{*}}}$.
For $b^{*}_{0}\in F^{*}$ and $x_{0}\in X_{0}$, we have

$\displaystyle\left\langle
b_{0}^{*},BTA(x_{0})\right\rangle_{(F_{0}^{*},F_{0})}$
$\displaystyle=\left\langle
b_{0}^{*},B(TAx_{0})\right\rangle_{(F_{0}^{*},F_{0})}=\left\langle
B^{*}b_{0}^{*},T(Ax_{0})\right\rangle_{(F^{*},F)}$ $\displaystyle=\left\langle
T^{\\#}_{|_{{F}^{*}}}B^{*}b_{0}^{*},Ax_{0}\right\rangle_{(X^{\\#},X)}=\left\langle
A^{\\#}T^{\\#}_{|_{{F}^{*}}}B^{*}b_{0}^{*},x_{0}\right\rangle_{(X_{0}^{\\#},X_{0})}.$

But also $\left\langle
b_{0}^{*},BTA(x)\right\rangle_{(F_{0}^{*},F_{0})}=\left\langle\left(BTA\right)^{\\#}_{|_{F^{*}_{0}}}b_{0}^{*},x_{0}\right\rangle_{(X_{0}^{\\#},X_{0})}$.
Therefore
$\left(BTA\right)^{\\#}_{|_{F^{*}_{0}}}=A^{\\#}T^{\\#}_{|_{{F}^{*}}}B^{*}$. ∎

######  Proposition 9.

$\mathfrak{A}^{L}_{dual}$ is a nonlinear ideal.

###### Proof.

The algebraic condition $\bf(\widetilde{PNOI_{0}})$ is satisfied, from Lemma 1
we obtain $\left(g\boxdot e\right)^{\\#}_{|_{{F}^{*}}}=\hat{e}\otimes
g\in\mathfrak{A}(F^{*},X^{\\#})$. To prove the algebraic condition
$\bf(\widetilde{PNOI_{1}})$, let $T$ and $S$ in
$\mathfrak{A}^{L}_{dual}(X,F)$. Let $T^{\\#}_{|_{{F}^{*}}}$ and
$S^{\\#}_{|_{{F}^{*}}}$ in $\mathfrak{A}(F^{*},X^{\\#})$, from Lemma 2 we have
$\left(T+S\right)^{\\#}_{|_{{F}^{*}}}=T^{\\#}_{|_{{F}^{*}}}+S^{\\#}_{|_{{F}^{*}}}\in\mathfrak{A}(F^{*},X^{\\#})$.
Let $A\in\operatorname{Lip}(X_{0},X)$, $T\in\mathfrak{A}^{L}_{dual}(X,F)$, and
$B\in\mathfrak{L}(F,F_{0})$. Also from Lemma 2 we have
$\left(BTA\right)^{\\#}_{|_{F^{*}_{0}}}=A^{\\#}T^{\\#}_{|_{{F}^{*}}}B^{*}\in\mathfrak{A}(F_{0}^{*},X_{0}^{\\#})$,
hence the algebraic condition $\bf(\widetilde{PNOI_{2}})$ is satisfied. ∎

The following proposition is obvious.

######  Proposition 10.

The rule

$dual:\mathfrak{A}\longrightarrow\left(\mathfrak{A}\right)^{L}_{dual}$

is a monotone Lipschitz procedure.

######  Proposition 11.

Let $\textfrak{F}^{L}$ be a nonlinear ideal of Lipschitz finite rank
operators, $\mathfrak{F}$ be an ideal of finite rank operators and
$(\mathfrak{F})^{L}_{dual}$ be a semi-Lipschitz procedure. Then
$\textfrak{F}^{L}=(\mathfrak{F})^{L}_{dual}$.

###### Proof.

Let $T\in\textfrak{F}^{L}(X,F)$, then $T$ can be represented in the form
$\sum\limits_{j=1}^{m}g_{j}\boxdot e_{j}$. From Lemma 1 and $E^{*}\otimes
F\equiv\mathfrak{F}(E,F)$ we have
$T^{\\#}_{|_{{F}^{*}}}=\sum\limits_{j=1}^{m}\hat{e}_{j}\otimes g_{j}\in
F^{**}\otimes X^{\\#}\equiv\mathfrak{F}(F^{*},X^{\\#})$. Hence
$T\in\mathfrak{F}^{L}_{dual}(X,F)$.

Let $T\in\mathfrak{F}^{L}_{dual}(X,F)$ then
$T^{\\#}_{|_{{F}^{*}}}\in\mathfrak{F}(F^{*},X^{\\#})$ hence
$T^{\\#}_{|_{{F}^{*}}}$ can be represented in the form
$\sum\limits_{j=1}^{m}\hat{e}_{j}\otimes g_{j}$. For $b^{*}\in F^{*}$ and
$x\in X$, we have

$\displaystyle\left\langle b^{*},Tx\right\rangle_{(F^{*},F)}$
$\displaystyle=\left\langle
T^{\\#}_{|_{{F}^{*}}}b^{*},x\right\rangle_{(X^{\\#},X)}=\left\langle\sum\limits_{j=1}^{m}\hat{e}_{j}\otimes
g_{j}\
(b^{*}),x\right\rangle_{(X^{\\#},X)}=\left\langle\sum\limits_{j=1}^{m}\hat{e}_{j}(b^{*})\cdot
g_{j}\ ,x\right\rangle_{(X^{\\#},X)}$
$\displaystyle=\left\langle\sum\limits_{j=1}^{m}b^{*}({e}_{j})\cdot g_{j}\
,x\right\rangle_{(X^{\\#},X)}=\sum\limits_{j=1}^{m}g_{j}(x)\cdot
b^{*}(e_{j})=\left\langle b^{*},\sum\limits_{j=1}^{m}g_{j}\boxdot e_{j}\
(x)\right\rangle_{(F^{*},F)}.$

Hence $T=\sum\limits_{j=1}^{m}g_{j}\boxdot e_{j}\in\textfrak{F}^{L}(X,F)$. ∎

####  3.1.4 Injective Nonlinear Ideals

Let $\textfrak{A}^{L}$ be a nonlinear ideal. A Lipschitz operator
$T\in\operatorname{Lip}(X,F)$ belongs to the injective hull
$\textfrak{A}^{L}_{inj}$ if $J_{F}T\in\textfrak{A}^{L}(X,F^{inj})$.

######  Proposition 12.

$\textfrak{A}^{L}_{inj}$ is a nonlinear ideal.

###### Proof.

The algebraic condition $\bf(\widetilde{PNOI_{0}})$ is satisfied, since
$g\boxdot e\in\textfrak{A}^{L}(X,F)$ and using nonlinear composition ideal
property we have $J_{F}(g\boxdot e)\in\textfrak{A}^{L}(X,F^{inj})$. To prove
the algebraic condition $\bf(\widetilde{PNOI_{1}})$, let $T$ and $S$ in
$\textfrak{A}^{L}_{inj}(X,F)$. Then $J_{F}T$ and $J_{F}S$ in
$\textfrak{A}^{L}(X,F^{inj})$, we have
$J_{F}(T+S)=J_{F}T+J_{F}S\in\textfrak{A}^{L}(X,F^{inj})$. Let
$A\in\operatorname{Lip}(X_{0},X)$, $T\in\textfrak{A}^{L}_{inj}(X,F)$, and
$B\in\mathfrak{L}(F,F_{0})$. Since $F_{0}^{inj}$ has the extension property,
there exists $B^{inj}\in\mathfrak{L}(F^{inj},F_{0}^{inj})$ such that

${X}$${F}$${F^{inj}}$${X_{0}}$${F_{0}}$${F_{0}^{inj}}$$\scriptstyle{T}$$\scriptstyle{J_{F}}$$\scriptstyle{B}$$\scriptstyle{B^{inj}}$$\scriptstyle{BTA}$$\scriptstyle{A}$$\scriptstyle{J_{F_{0}}}$

Consequently
$J_{F_{0}}\left(BTA\right)=B^{inj}\left(J_{F}T\right)A\in\textfrak{A}^{L}$,
hence the algebraic condition $\bf(\widetilde{PNOI_{2}})$ is satisfied. ∎

######  Proposition 13.

The rule

$\text{inj}:\textfrak{A}^{L}\longrightarrow\textfrak{A}^{L}_{inj}$

is a hull Lipschitz procedure.

###### Proof.

The property $\bf(M^{\prime})$ is obvious. To show the idempotence, let
$T\in\operatorname{Lip}(X,F)$ belong to
$\left(\textfrak{A}^{L}_{inj}\right)_{inj}$. Then
$J_{F}T\in\textfrak{A}^{L}_{inj}(X,F^{inj})$, and the preceding lemma implies
$J_{F}T\in\textfrak{A}^{L}(X,F^{inj})$. Consequently
$T\in\textfrak{A}^{L}_{inj}(X,F)$. Thus
$\left(\textfrak{A}^{L}_{inj}\right)_{inj}\subseteq\textfrak{A}^{L}_{inj}$.
The converse inclusion is trivial. ∎

######  Lemma 3.

Let $F$ be a Banach space possessing the extension property. Then
$\textfrak{A}^{L}(X,F)=\textfrak{A}^{L}_{inj}(X,F)$.

###### Proof.

By hypothesis there exists $B\in\mathfrak{L}(F^{inj},F)$ such that
$BJ_{F}=I_{F}$. Therefore $T\in\textfrak{A}^{L}_{inj}(X,F)$ implies that
$T=B\left(J_{F}T\right)\in\textfrak{A}^{L}(X,F)$. This proves that
$\textfrak{A}^{L}_{inj}\subseteq\textfrak{A}^{L}$. The converse inclusion is
obvious. ∎

######  Proposition 14.

The rule

$inj:\textfrak{A}^{L}\longrightarrow\textfrak{A}^{L}_{inj}$

is a hull Lipschitz procedure.

###### Proof.

The property $\bf(M^{\prime})$ is obvious. To show the idempotence, let
$T\in\operatorname{Lip}(X,F)$ belong to
$\left(\textfrak{A}^{L}_{inj}\right)_{inj}$. Then
$J_{F}T\in\textfrak{A}^{L}_{inj}(X,F^{inj})$, and the preceding lemma implies
$J_{F}T\in\textfrak{A}^{L}(X,F^{inj})$. Consequently
$T\in\textfrak{A}^{L}_{inj}(X,F)$. Thus
$\left(\textfrak{A}^{L}_{inj}\right)_{inj}\subseteq\textfrak{A}^{L}_{inj}$.
The converse inclusion is trivial. ∎

###  3.2 Minimal Nonlinear Ideals

Let $\mathfrak{A}$ be an ideal. A Lipschitz operator
$T\in\operatorname{Lip}(X,F)$ belongs to the associated minimal ideal
$(\mathfrak{A})^{L}_{min}$ if $T=BT_{0}A$, where $B\in\mathfrak{G}(F_{0},F)$,
$T_{0}\in\mathfrak{A}(G_{0},F_{0})$, and $A\in\textfrak{G}^{L}(X,G_{0})$. In
the other words
$(\mathfrak{A})^{L}_{min}:=\mathfrak{G}\circ\mathfrak{A}\circ\textfrak{G}^{L}$,
where $\mathfrak{G}$ be an ideal of approximable operators between arbitrary
Banach spaces.

######  Proposition 15.

$(\mathfrak{A})^{L}_{min}$ is a nonlinear ideal.

###### Proof.

The algebraic condition $\bf(\widetilde{PNOI_{0}})$ is satisfied, since the
elementary Lipschitz tensor $g\boxdot e$ admits a factorization

$g\boxdot e:X\stackrel{{\scriptstyle g\boxdot
1}}{{\longrightarrow}}\mathbb{K}\stackrel{{\scriptstyle 1\otimes
1}}{{\longrightarrow}}\mathbb{K}\stackrel{{\scriptstyle 1\otimes
e}}{{\longrightarrow}}F,$

where $1\otimes e\in\mathfrak{G}\left(\mathbb{K},F\right)$, $1\otimes
1\in\mathfrak{A}\left(\mathbb{K},\mathbb{K}\right)$, and $g\boxdot
1\in\textfrak{G}^{L}\left(X,\mathbb{K}\right)$. To prove the algebraic
condition $\bf(\widetilde{PNOI_{1}})$, let
$T_{i}\in\mathfrak{G}\circ\mathfrak{A}\circ\textfrak{G}^{L}(X,F)$. Then
$T_{i}=B_{i}T_{0}^{i}A_{i}$, where $B_{i}\in\mathfrak{G}(F_{0}^{i},F)$,
$T_{0}^{i}\in\mathfrak{A}(G_{0}^{i},F_{0}^{i})$, and
$A_{i}\in\textfrak{G}^{L}(X,G_{0}^{i})$. Put $B:=B_{1}\circ Q_{1}+B_{2}\circ
Q_{2}$, $T_{0}:=\tilde{J}_{1}\circ
T_{0}^{1}\circ\tilde{Q}_{1}+\tilde{J}_{2}\circ T_{0}^{2}\circ\tilde{Q}_{2}$,
and $A:=J_{1}\circ A_{1}+J_{2}\circ A_{2}$. Now $T_{1}+T_{2}=B\circ T_{0}\circ
A$, $B\in\mathfrak{G}(F_{0},F)$, $T_{0}\in\mathfrak{A}(G_{0},F_{0})$, and
$A\in\textfrak{G}^{L}(X,G_{0})$ imply
$T_{1}+T_{2}\in\mathfrak{G}\circ\mathfrak{A}\circ\textfrak{G}^{L}(X,F)$. Let
$A\in\operatorname{Lip}(X_{0},X)$,
$T\in\mathfrak{G}\circ\mathfrak{A}\circ\textfrak{G}^{L}(X,F)$, and
$B\in\mathfrak{L}(F,R_{0})$. Then $T$ admits a factorization

$T:X\stackrel{{\scriptstyle\widetilde{A}}}{{\longrightarrow}}G_{0}\stackrel{{\scriptstyle
T_{0}}}{{\longrightarrow}}F_{0}\stackrel{{\scriptstyle\widetilde{B}}}{{\longrightarrow}}F,$

where $\widetilde{B}\in\mathfrak{G}(F_{0},F)$,
$T_{0}\in\mathfrak{A}(G_{0},F_{0})$, and
$\widetilde{A}\in\textfrak{G}^{L}(X,G_{0})$. To show that
$BTA\in\mathfrak{G}\circ\mathfrak{A}\circ\textfrak{G}^{L}(X_{0},R_{0})$. By
using the linear and nonlinear composition ideal properties, we obtain
$B\circ\widetilde{B}\in\mathfrak{G}\left(F_{0},R_{0}\right)$ and
$\widetilde{A}\circ A\in\textfrak{G}^{L}\left(X_{0},G_{0}\right)$. Hence the
Lipschitz operator $BTA$ admits a factorization

$BTA:X_{0}\stackrel{{\scriptstyle\widetilde{\widetilde{A\,}}}}{{\longrightarrow}}G_{0}\stackrel{{\scriptstyle
T_{0}}}{{\longrightarrow}}F_{0}\stackrel{{\scriptstyle\widetilde{\widetilde{B\,}}}}{{\longrightarrow}}R_{0},$

where $\widetilde{\widetilde{B\,}}=B\circ\widetilde{B}$ and
$\widetilde{\widetilde{A\,}}=\widetilde{A}\circ A$, hence the algebraic
condition $\bf(\widetilde{PNOI_{2}})$ is satisfied. ∎

######  Proposition 16.

The rule

$min:\mathfrak{A}\longrightarrow(\mathfrak{A})^{L}_{min}$

is a monotone Lipschitz procedure.

######  Remark 4.

  1. $\bf(1)$

It is evident $(\mathfrak{A})^{L}_{min}\subseteq\textfrak{G}^{L}$.

  2. $\bf(2)$

If $\textfrak{A}^{L}$ is a closed nonlinear ideal, then
$(\mathfrak{A})^{L}_{min}\subseteq\textfrak{A}^{L}$.

######  Proposition 17.

Let $\mathfrak{A}$ be a closed ideal. Then
$(\mathfrak{A})^{L}_{min}=\textfrak{G}^{L}$. In particular, the linear and
nonlinear ideals of approximable operators are related by
$(\mathfrak{G})^{L}_{min}=\textfrak{G}^{L}$.

The prove of the counterpart of this proposition for ideals of linear
operators requires the notion of idempotence of ideals, see [14, Prop. 4.8.4].
In particular, the equalities

$\mathfrak{F}\circ\mathfrak{F}=\mathfrak{F}\ \ \ \text{and}\ \ \
\mathfrak{G}\circ\mathfrak{G}=\mathfrak{G}$ (9)

are needed. Since idempotence does not make sense for nonlinear ideals, we
instead use the following equalities.

######  Proposition 18.

Any Lipschitz finite operator can be written as a product of a linear operator
with finite rank and a Lipschitz finite operator. Any Lipschitz approximable
operator can be written as a product of a linear approximable operator and a
Lipschitz approximable operator. That is,
$\mathfrak{F}\circ\textfrak{F}^{L}=\textfrak{F}^{L}$ and
$\mathfrak{G}\circ\textfrak{G}^{L}=\textfrak{G}^{L}$.

###### Proof.

Let $T=\sum\limits_{j=1}^{m}g_{j}\boxdot e_{j}$ with $g_{1},\cdots,g_{m}$ in
$X^{\\#}$ and $e_{1},\cdots,e_{m}$ in $F$ be a Lipschitz finite operator. Let
$F_{0}$ be the finite dimensional subspace of $F$ spanned by
$e_{1},\cdots,e_{m}$ and let $J:F_{0}\to F$ be the embedding. Obviously, $J$
is a linear operator with finite rank. Moreover, let $T_{0}$ be the operator
$T$ considered as an operator from $X$ to $F_{0}$. Then $T=JT_{0}$ is the
required factorization. Observe that we also have
$\operatorname{Lip}(T_{0})\|J\|=\operatorname{Lip}(T)$. The inclusion
$\mathfrak{F}\circ\textfrak{F}^{L}\subseteq\textfrak{F}^{L}$ is obvious.

Now let $T\in\textfrak{G}^{L}(X,F)$. Since $T$ can be approximated by
Lipschitz finite operators, we can also find Lipschitz finite operators
$T_{n}\in\textfrak{F}^{L}(X,F)$ such that the sum
$T=\sum\limits_{n=1}^{\infty}T_{n}$ converges absolutely in
$\operatorname{Lip}(X,F)$, i.e.
$\sum\limits_{n=1}^{\infty}\operatorname{Lip}(T_{n})<\infty$. Now each $T_{n}$
can be factored as $T_{n}=V_{n}U_{n}$ with $V_{n}\in\mathfrak{F}(F_{i},F)$ and
$U_{n}\in\textfrak{F}^{L}(X,F_{i})$ such that $F_{i}$ is a suitable Banach
space and $\|V_{n}\|\operatorname{Lip}(U_{n})=\operatorname{Lip}(T)$. By
homogeneity, we may assume that
$\|V_{n}\|^{2}=\operatorname{Lip}(U_{n})^{2}=\operatorname{Lip}(T_{n})$. Let
$M:=\ell_{2}(F_{n})$ and put

$V:=\sum\limits_{n=1}^{\infty}J_{n}V_{n}\ \ \ \text{and}\ \ \
U:=\sum\limits_{n=1}^{\infty}U_{n}Q_{n}.$

Then

$\operatorname{Lip}(U)^{2}\leq\sum\limits_{n=1}^{\infty}\operatorname{Lip}(U_{n})^{2}<\infty\
\ \ \text{and}\ \ \
\left\|V\right\|^{2}\leq\sum\limits_{n=1}^{\infty}\left\|V_{n}\right\|^{2}<\infty.$

This shows that the series defining $U$ and $V$ are absolutely convergent in
$\operatorname{Lip}(X,M)$ and $\mathfrak{L}(M,F)$, respectively. Hence
$U\in\textfrak{G}^{L}(X,M)$ and $V\in\mathfrak{G}(M,F)$ and $T=VU$ is the
required factorization. Again,
$\mathfrak{G}\circ\textfrak{G}^{L}\subseteq\textfrak{G}^{L}$ is obvious. ∎

We can now prove Proposition 17.

###### Proof of Proposition 17.

By (9) and Proposition 18, we have

$(\mathfrak{G})^{L}_{min}=\mathfrak{G}\circ\mathfrak{G}\circ\textfrak{G}^{L}=\mathfrak{G}\circ\textfrak{G}^{L}=\textfrak{G}^{L}.$

If now $\mathfrak{A}$ is a closed ideal, then
$\mathfrak{G}\subseteq\mathfrak{A}$ implies

$\textfrak{G}^{L}=(\mathfrak{G})^{L}_{min}=\mathfrak{G}\circ\mathfrak{G}\circ\textfrak{G}^{L}\subseteq\mathfrak{G}\circ\mathfrak{A}\circ\textfrak{G}^{L}=(\mathfrak{A})^{L}_{min}.$

The reverse implication was already observed in Remark 4. ∎

###  3.3 Lipschitz interpolative ideal procedure between metric spaces and
Banach spaces

######  Proposition 19.

$\left[\textfrak{A}^{L}_{\text{inj}},\mathbf{A}^{L}_{\text{inj}}\right]$ is a
$p$-normed nonlinear ideal.

###### Proof.

By Proposition 12 the algebraic conditions of Definition 1 are hold. Then the
injective hull $\textfrak{A}^{L}_{inj}$ is a nonlinear ideal. To prove the
norm condition $\bf(\widetilde{PNOI_{1}})$, let $T$ and $S$ in
$\textfrak{A}^{L}_{\text{inj}}(X,F)$. Then

$\displaystyle\mathbf{A}^{L}_{\text{inj}}\left(S+T\right)^{p}$
$\displaystyle:=\mathbf{A}^{L}\left[J_{F}\left(S+T\right)\right]^{p}=\mathbf{A}^{L}\left[J_{F}S+J_{F}T\right]^{p}$
$\displaystyle\leq\mathbf{A}^{L}\left(J_{F}S\right)^{p}+\mathbf{A}^{L}\left(J_{F}T\right)^{p}$
$\displaystyle=\mathbf{A}^{L}_{\text{inj}}(S)^{p}+\mathbf{A}^{L}_{\text{inj}}(T)^{p}.$

Let $A\in\operatorname{Lip}(X_{0},X)$,
$T\in\textfrak{A}^{L}_{\text{inj}}(X,F)$, and $B\in\mathfrak{L}(F,F_{0})$.
Then

$\displaystyle\mathbf{A}^{L}_{\text{inj}}\left(BTA\right)$
$\displaystyle:=\mathbf{A}^{L}\left(J_{F}\left(BTA\right)\right)=\mathbf{A}^{L}\left(B^{\text{inj}}\left(J_{F}T\right)A\right)$
$\displaystyle\leq\left\|B^{\text{inj}}\right\|\mathbf{A}^{L}\left(J_{F}T\right)\operatorname{Lip}(A)$
$\displaystyle=\left\|B\right\|\mathbf{A}^{L}_{\text{inj}}(T)\operatorname{Lip}(A).$

Hence the norm condition $\bf(\widetilde{PNOI_{2}})$ is satisfied. ∎

######  Definition 3.

Let $0\leq\theta<1$ and $\left[\textfrak{A}^{L},\mathbf{A}^{L}\right]$ be a
normed nonlinear ideal. A Lipschitz operator $T$ from $X$ into $F$ belongs to
$\textfrak{A}^{L}_{\theta}(X,F)$ if there exist a Banach space $G$ and a
Lipschitz operator $S\in\textfrak{A}^{L}(X,G)$ such that

$\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|\leq\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|^{1-\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X.$ (10)

For each $T\in\textfrak{A}^{L}_{\theta}(X,F)$, we set

$\mathbf{A}^{L}_{\theta}(T):=\inf\mathbf{A}^{L}(S)^{1-\theta}$ (11)

where the infimum is taken over all Lipschitz operators $S$ admitted in (10).

Note that $\operatorname{Lip}(T)\leq\mathbf{A}^{L}_{\theta}(T)$, by
definition.

######  Proposition 20.

$\left[\textfrak{A}^{L}_{\theta},\mathbf{A}^{L}_{\theta}\right]$ is an
injective Banach nonlinear ideal.

###### Proof.

The condition of injective hull and the algebraic conditions of Definition 1
are not difficult to prove it. Let $x^{\prime},\;x^{\prime\prime}\in X$ and
$e\in F$ the norm condition $\bf(\widetilde{PNOI_{0}})$ is satisfied. Indeed

$\left\|(g\boxdot e)x^{\prime}-(g\boxdot
e)x^{\prime\prime}|F\right\|\leq\left\|\left(\operatorname{Lip}(g)\cdot\left\|e\right\|\right)^{\frac{\theta}{1-\theta}}\left[(g\boxdot
e)x^{\prime}-(g\boxdot e)x^{\prime\prime}\right]|F\right\|^{1-\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}.$ (12)

Since a Lipschitz operator
$S:=(\operatorname{Lip}(g)\cdot\left\|e\right\|)^{\frac{\theta}{1-\theta}}(g\boxdot
e)\in\textfrak{A}^{L}(X,F)$, hence $g\boxdot
e\in\textfrak{A}^{L}_{\theta}(X,F)$. From (11) we have

$\displaystyle\mathbf{A}^{L}_{\theta}(g\boxdot e)$
$\displaystyle:=\inf\mathbf{A}^{L}\big{(}(\operatorname{Lip}(g)\cdot\left\|e\right\|)^{\frac{\theta}{1-\theta}}g\boxdot
e\big{)}^{1-\theta}$
$\displaystyle\leq(\operatorname{Lip}(g)\cdot\left\|e\right\|)^{\theta}\mathbf{A}^{L}(g\boxdot
e\big{)}^{1-\theta}$
$\displaystyle=(\operatorname{Lip}(g)\cdot\left\|e\right\|)^{\theta}\cdot(\operatorname{Lip}(g)\cdot\left\|e\right\|)^{1-\theta}$
$\displaystyle=\operatorname{Lip}(g)\cdot\left\|e\right\|.$

The converse inequality is obvious. To prove the norm condition
$\bf(\widetilde{PNOI_{1}})$, let $T_{1}$ and $T_{2}$ in
$\textfrak{A}^{L}_{\theta}(X,F)$. Given $\epsilon>0$, there is a Banach space
$G_{i}$ and a Lipschitz operator $S_{i}\in\textfrak{A}^{L}(X,G_{i})$, $i=1,2$
such that

$\left\|T_{i}x^{\prime}-T_{i}x^{\prime\prime}|F\right\|\leq\left\|\mathbf{A}^{L}(S_{i})^{-\theta}\left(S_{i}x^{\prime}-S_{i}x^{\prime\prime}\right)|G\right\|^{1-\theta}\left(\mathbf{A}^{L}(S_{i})^{1-\theta}\right)^{\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X$ (13)

and
$\mathbf{A}^{L}(S_{i})^{1-\theta}\leq(1+\epsilon)\cdot\mathbf{A}^{L}_{\theta}(T_{i})\
\left(i=1,2\right)$. Introducing the $\ell_{1}$-sum $G:=G_{1}\oplus G_{2}$ and
the Lipschitz operator
$S:=\mathbf{A}^{L}(S_{1})^{-\theta}J_{1}S_{1}+\mathbf{A}^{L}(S_{2})^{-\theta}J_{2}S_{2}\in\textfrak{A}^{L}(X,G)\
\left(J_{1},J_{2}\ \text{the canonical injections}\right)$ and applying the
Hölder inequality, we get for all $x^{\prime},x^{\prime\prime}\in X$

$\displaystyle\left\|(T_{1}+T_{2})x^{\prime}-(T_{1}+T_{2})x^{\prime\prime}|F\right\|$
$\displaystyle\leq\left\|T_{1}x^{\prime}+T_{1}x^{\prime\prime}|F\right\|+\left\|T_{1}x^{\prime}+T_{1}x^{\prime\prime}|F\right\|$
$\displaystyle\leq\sum\limits_{i=1}^{2}\left\|\mathbf{A}^{L}(S_{i})^{-\theta}\left(S_{i}x^{\prime}-S_{i}x^{\prime\prime}\right)|G_{i}\right\|^{1-\theta}\left(\mathbf{A}^{L}(S_{i})^{1-\theta}\right)^{\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}$
$\displaystyle\leq\left[\sum\limits_{i=1}^{2}\left\|\mathbf{A}^{L}(S_{i})^{-\theta}\left(S_{i}x^{\prime}-S_{i}x^{\prime\prime}\right)|G_{i}\right\|\right]^{1-\theta}\left(\sum\limits_{i=1}^{2}\mathbf{A}^{L}(S_{i})^{1-\theta}\right)^{\theta}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}$
$\displaystyle=\left(\mathbf{A}^{L}(S_{1})^{1-\theta}+\mathbf{A}^{L}(S_{2})^{1-\theta}\right)^{\theta}\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|^{1-\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}.$

Hence $T_{1}+T_{2}\in\textfrak{A}^{L}_{\theta}(X,F)$ and furthermore, for
$p=1$ we have

$\displaystyle\mathbf{A}^{L}_{\theta}(T_{1}+T_{2})$
$\displaystyle\leq\left[\mathbf{A}^{L}(S_{1})^{1-\theta}+\mathbf{A}^{L}(S_{2})^{1-\theta}\right]^{\theta}\mathbf{A}^{L}(S)^{1-\theta}$
$\displaystyle\leq\mathbf{A}^{L}(S_{1})^{1-\theta}+\mathbf{A}^{L}(S_{2})^{1-\theta}$
$\displaystyle\leq(1+\epsilon)\cdot\left(\mathbf{A}^{L}_{\theta}(T_{1})+\mathbf{A}^{L}_{\theta}(T_{2})\right).$

To prove the norm condition $\bf(\widetilde{PNOI_{2}})$, let
$A\in\mathscr{L}(X_{0},X)$, $T\in\textfrak{A}^{L}_{\theta}(X,F)$, and
$B\in\mathfrak{L}(F,F_{0})$ and $x^{\prime}_{0},x^{\prime\prime}_{0}$ in $X$.
Then

$\displaystyle\left\|BTAx^{\prime}_{0}-BTAx^{\prime\prime}_{0}|F_{0}\right\|$
$\displaystyle\leq\left\|B\right\|\cdot\left\|TAx^{\prime}_{0}-TAx^{\prime\prime}_{0}|F\right\|$
$\displaystyle\leq\left\|B\right\|\cdot\left\|S(Ax^{\prime}_{0})-S(Ax^{\prime\prime}_{0})|G\right\|^{1-\theta}\cdot
d_{X}(Ax^{\prime}_{0},Ax^{\prime\prime}_{0})^{\theta}$
$\displaystyle\leq\left\|B\right\|\cdot\operatorname{Lip}(A)^{\theta}\cdot\left\|S\circ
A(x^{\prime}_{0})-S\circ
A(x^{\prime\prime}_{0})|G\right\|^{1-\theta}d_{X_{0}}(x^{\prime}_{0},x^{\prime\prime}_{0})^{\theta}$
$\displaystyle\leq\left\|\left\|B\right\|^{\frac{1}{1-\theta}}\cdot\operatorname{Lip}(A)^{\frac{\theta}{1-\theta}}\left(S\circ
A(x^{\prime}_{0})-S\circ
A(x^{\prime\prime}_{0})\right)|G\right\|^{1-\theta}d_{X_{0}}(x^{\prime}_{0},x^{\prime\prime}_{0})^{\theta}.$
(14)

Since a Lipschitz map
$\widetilde{S}:=\left(\left\|B\right\|^{\frac{1}{1-\theta}}\cdot\operatorname{Lip}(A)^{\frac{\theta}{1-\theta}}\right)S\circ
A\in\textfrak{A}^{L}(X_{0},G)$, hence
$BTA\in\textfrak{A}^{L}_{\theta}(X_{0},F_{0})$. Moreover, from (11) we have

$\displaystyle\mathbf{A}^{L}_{\theta}(BTA)$
$\displaystyle:=\inf\mathbf{A}^{L}(\widetilde{S})^{1-\theta}$
$\displaystyle\leq\mathbf{A}^{L}\left((\left\|B\right\|^{\frac{1}{1-\theta}}\cdot\operatorname{Lip}(A)^{\frac{\theta}{1-\theta}})S\circ
A\right)^{1-\theta}$
$\displaystyle\leq\left\|B\right\|\cdot\operatorname{Lip}(A)^{\theta}\cdot\mathbf{A}^{L}(S\circ
A)^{1-\theta}$
$\displaystyle=\left\|B\right\|\cdot\operatorname{Lip}(A)\cdot\mathbf{A}^{L}(S)^{1-\theta}.$
(15)

Taking the infimum over all such $S\in\textfrak{A}^{L}(X,G)$ on the right side
of (3.3), we have

$\mathbf{A}^{L}_{\theta}(BTA)\leq\left\|B\right\|\cdot\mathbf{A}^{L}_{\theta}(T)\cdot\operatorname{Lip}(A).$

To prove the completeness, condition $\bf(\widetilde{PNOI_{3}})$, let
$(T_{n})_{n\in\mathbb{N}}$ be a sequence of Lipschitz operator in
$\textfrak{A}^{L}_{\theta}(X,F)$ such that
$\sum\limits_{n=1}^{\infty}\mathbf{A}^{L}_{\theta}(T_{n})<\infty$. Since
$\operatorname{Lip}(T)\leq\mathbf{A}^{L}_{\theta}(T)$ and
$\operatorname{Lip}(X,F)$ is a Banach space, there exists
$T=\sum\limits_{n=1}^{\infty}T_{n}\in\operatorname{Lip}(X,F)$. Let
$S_{n}\in\textfrak{A}^{L}(X,G_{n})$ such that

$\left\|T_{n}x^{\prime}-T_{n}x^{\prime\prime}|F\right\|\leq\left\|S_{n}x^{\prime}-S_{n}x^{\prime\prime}|G_{n}\right\|^{1-\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X,$

and
$\mathbf{A}^{L}(S_{n})^{1-\theta}\leq\mathbf{A}^{L}_{\theta}(T_{n})+\frac{\epsilon}{2^{n}}$.
Then

$\left(\sum\limits_{n=1}^{\infty}\mathbf{A}^{L}(S_{n})\right)^{1-\theta}\leq\sum\limits_{n=1}^{\infty}\mathbf{A}^{L}(S_{n})^{1-\theta}\leq\sum\limits_{n=1}^{\infty}\mathbf{A}^{L}_{\theta}(T_{n})+\epsilon<\infty.$

Let $S=\sum\limits_{n=1}^{\infty}S_{n}\in\textfrak{A}^{L}(X,G)$, where $G$ is
the $\ell_{1}$-sum of all $G_{n}$. Hence

$\displaystyle\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|\leq\sum\limits_{n=1}^{\infty}\left\|T_{n}x^{\prime}-T_{n}x^{\prime\prime}|F_{n}\right\|$
$\displaystyle\leq\sum\limits_{n=1}^{\infty}\left\|S_{n}x^{\prime}-S_{n}x^{\prime\prime}|G_{n}\right\|^{1-\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}$
$\displaystyle\leq\left\|Sx^{\prime}-Sx^{\prime\prime}|G_{n}\right\|^{1-\theta}\left(\sum\limits_{n=1}^{\infty}\mathbf{A}^{L}(S_{n})^{1-\theta}\right)^{\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}.$

This implies that $T\in\textfrak{A}^{L}_{\theta}(X,F)$ and

$\mathbf{A}^{L}_{\theta}(T)\leq\sum\limits_{n=1}^{\infty}\mathbf{A}^{L}(S_{n})^{1-\theta}\leq\sum\limits_{n=1}^{\infty}\mathbf{A}^{L}_{\theta}(T_{n})+\epsilon<\infty.$

We have

$\mathbf{A}^{L}_{\theta}\left(T-\sum\limits_{j=1}^{n}T_{j}\right)=\mathbf{A}^{L}_{\theta}\left(\sum\limits_{k=n+1}^{\infty}T_{k}\right)\leq\sum\limits_{k=n+1}^{\infty}\mathbf{A}^{L}_{\theta}(S_{k})^{1-\theta}.$

Thus, $T=\sum\limits_{n=1}^{\infty}T_{n}$. ∎

######  Remark 5.

If $\theta=0$, then the nonlinear ideal $\textfrak{A}^{L}_{\theta}$ is just
the injective hull of nonlinear ideal $\textfrak{A}^{L}$ and Lipschitz norms
are equal. Further properties are given in.

######  Proposition 21.

Let $0\leq\theta,\theta_{1},\theta_{2}<1$. Then the following holds.

  1. $\bf(a)$

$\textfrak{A}^{L}_{\theta_{1}}\subset\textfrak{A}^{L}_{\theta_{2}}$ if
$\theta_{1}\leq\theta_{2}$.

  2. $\bf(b)$

$\textfrak{A}^{L}_{\text{inj}}\subset\textfrak{A}^{L}_{\theta}$.

  3. $\bf(c)$

$\left(\textfrak{A}^{L}_{\theta_{1}}\right)_{\theta_{2}}\subset\textfrak{A}^{L}_{\theta_{1}+\theta_{2}-\theta_{1}\theta_{2}}$.

###### Proof.

To verify $\bf(a)$, let $T\in\textfrak{A}^{L}_{\theta_{1}}(X,F)$ and
$\epsilon>0$. Then

$\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|\leq\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|^{1-\theta_{1}}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta_{1}},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X,$

holds for a suitable Banach space $G$ and a Lipschitz operator
$S\in\textfrak{A}^{L}(X,G)$ with
$\mathbf{A}^{L}(S)^{1-\theta_{1}}\leq(1+\epsilon)\cdot\mathbf{A}^{L}_{\theta_{1}}(T)$.
Since

$\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|\leq\operatorname{Lip}(S)^{\theta_{2}-\theta_{1}}\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|^{1-\theta_{2}}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta_{2}},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X,$

Since a Lipschitz map
$\widetilde{S}:=\operatorname{Lip}(S)^{\frac{\theta_{2}-\theta_{1}}{1-\theta_{2}}}S\in\textfrak{A}^{L}(X,G)$,
hence $T\in\textfrak{A}^{L}_{\theta_{2}}(X,F)$ and

$\mathbf{A}^{L}_{\theta_{2}}(T)\leq\mathbf{A}^{L}(\widetilde{S})^{1-\theta_{2}}\leq\operatorname{Lip}(S)^{\theta_{2}-\theta_{1}}\mathbf{A}^{L}(S)^{1-\theta_{2}}\leq\mathbf{A}^{L}(S)^{1-\theta_{1}}\leq(1+\epsilon)\cdot\mathbf{A}^{L}_{\theta_{1}}(T).$

To verify $\bf(b)$, let $T\in\textfrak{A}^{L}_{\text{inj}}(X,F)$. Then
$J_{F}T\in\textfrak{A}^{L}(X,F^{\text{inj}})$ and

$\displaystyle\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|$
$\displaystyle=\left\|J_{F}(Tx^{\prime})-J_{F}(Tx^{\prime\prime})|F^{\text{inj}}\right\|$
$\displaystyle\leq\operatorname{Lip}(T)^{\theta}\cdot\left\|J_{F}\circ
T(x^{\prime})-J_{F}\circ
T(x^{\prime\prime})|F^{\text{inj}}\right\|^{1-\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}.$

Since $G:=F^{\text{inj}}$ and a Lipschitz map
$S:=\operatorname{Lip}(T)^{\frac{\theta}{1-\theta}}J_{F}\circ
T\in\textfrak{A}^{L}(X,G)$, hence $T\in\textfrak{A}^{L}_{\theta}(X,F)$.
Moreover,

$\displaystyle\mathbf{A}^{L}_{\theta}(T):=\inf\mathbf{A}^{L}(S)^{1-\theta}$
$\displaystyle\leq\mathbf{A}^{L}(\operatorname{Lip}(T)^{\frac{\theta}{1-\theta}}J_{F}\circ
T)^{1-\theta}$
$\displaystyle\leq\operatorname{Lip}(T)^{\theta}\cdot\mathbf{A}^{L}(J_{F}\circ
T)^{1-\theta}$
$\displaystyle:=\operatorname{Lip}(T)^{\theta}\cdot\mathbf{A}_{\text{inj}}^{L}(T)^{1-\theta}$
$\displaystyle\leq\mathbf{A}_{\text{inj}}^{L}(T)^{\theta}\cdot\mathbf{A}_{\text{inj}}^{L}(T)^{1-\theta}$
$\displaystyle=\mathbf{A}_{\text{inj}}^{L}(T).$

To verify $\bf(c)$, let
$T\in\left(\textfrak{A}^{L}_{\theta_{1}}\right)_{\theta_{2}}(X,F)$ and
$\epsilon>0$. Then

$\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|\leq\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|^{1-\theta_{2}}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta_{2}},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X,$ (16)

holds for a suitable Banach space $G$ and a Lipschitz operator
$S\in\textfrak{A}_{\theta_{1}}^{L}(X,G)$ with
$\mathbf{A}_{\theta_{1}}^{L}(S)^{1-\theta_{2}}\leq(1+\epsilon)\cdot\left(\mathbf{A}_{\theta_{1}}^{L}(T)\right)_{\theta_{2}}$
and

$\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|\leq\left\|Rx^{\prime}-Rx^{\prime\prime}|G\right\|^{1-\theta_{1}}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta_{1}},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X,$ (17)

holds for a suitable Banach space $\widetilde{G}$ and a Lipschitz operator
$R\in\textfrak{A}^{L}(X,\widetilde{G})$ with
$\mathbf{A}^{L}(R)^{1-\theta_{1}}\leq(1+\epsilon)\cdot\mathbf{A}_{\theta_{1}}^{L}(S)$.
From (16) and (17) we have

$\displaystyle\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|$
$\displaystyle\leq\left\|Rx^{\prime}-Rx^{\prime\prime}|\widetilde{G}\right\|^{({1-\theta_{1}})\cdot({1-\theta_{2}})}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta_{1}\cdot(1-\theta_{2})}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta_{2}}$
$\displaystyle\leq\left\|Rx^{\prime}-Rx^{\prime\prime}|\widetilde{G}\right\|^{1-\theta_{2}-\theta_{1}+\theta_{1}\cdot\theta_{2}}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta_{2}+\theta_{1}-\theta_{1}\cdot\theta_{2}},$

hence
$T\in\textfrak{A}^{L}_{\theta_{1}+\theta_{2}-\theta_{1}\theta_{2}}(X,F)$.
Moreover,

$\displaystyle\mathbf{A}^{L}_{\theta_{1}+\theta_{2}-\theta_{1}\theta_{2}}(T)$
$\displaystyle\leq\mathbf{A}^{L}(R)^{1-\theta_{1}-\theta_{2}+\theta_{1}\theta_{2}}$
$\displaystyle\leq(1+\epsilon)^{\frac{1-\theta_{1}-\theta_{2}+\theta_{1}\theta_{2}}{1-\theta_{1}}}\cdot\mathbf{A}_{\theta_{1}}^{L}(S)^{\frac{1-\theta_{1}-\theta_{2}+\theta_{1}\theta_{2}}{1-\theta_{1}}}$
$\displaystyle\leq(1+\epsilon)^{\frac{1-\theta_{1}-\theta_{2}+\theta_{1}\theta_{2}}{1-\theta_{1}}}\cdot(1+\epsilon)^{\frac{1}{1-\theta_{2}}\cdot\frac{1-\theta_{1}-\theta_{2}+\theta_{1}\theta_{2}}{1-\theta_{1}}}\cdot\left(\mathbf{A}_{\theta_{1}}^{L}(T)\right)_{\theta_{2}}^{\frac{1}{1-\theta_{2}}\cdot\frac{1-\theta_{1}-\theta_{2}+\theta_{1}\theta_{2}}{1-\theta_{1}}}$
$\displaystyle=(1+\epsilon)^{\frac{2-2\theta_{1}-\theta_{2}+\theta_{1}\theta_{2}}{1-\theta_{1}}}\cdot\left(\mathbf{A}_{\theta_{1}}^{L}(T)\right)_{\theta_{2}}.$

∎

###  3.4 Lipschitz $\left(p,\theta,q,\nu\right)$-dominated operators

$1\leq p,q<\infty$ and $0\leq\theta,\nu<1$ such that
$\frac{1}{r}+\frac{1-\theta}{p}+\frac{1-\nu}{q}=1$ with $1\leq r<\infty$. We
then introduce the following definition.

######  Definition 4.

A Lipschitz operator $T$ from $X$ to $F$ is called Lipschitz
$\left(p,\theta,q,\nu\right)$-dominated if there exists a Banach spaces $G$
and $H$, a Lipschitz operator $S\in\Pi_{p}^{L}(X,G)$, a bounded operator
$R\in\Pi_{q}(F^{*},H)$ and a positive constant $C$ such that

$\left|\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|\leq C\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|R(b^{*})|H\right\|^{1-\nu}$
(18)

for arbitrary finite sequences $x^{\prime}$, $x^{\prime\prime}$ in $X$, and
$b^{*}\subset F^{*}$.

Let us denote by $\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$ the
class of all Lipschitz $\left(p,\theta,q,\nu\right)$-dominated operators from
$X$ to $F$ with

$D_{\left(p,\theta,q,\nu\right)}^{L}(T)=\inf\left\\{C\cdot\pi_{p}^{L}(S)^{1-\theta}\cdot\pi_{q}(R)^{1-\nu}\right\\},$

where the infimum is taken over all Lipschitz operator $S$, bounded operator
$R$, and constant $C$ admitted in (18).

######  Proposition 22.

The ordered pair
$\left(\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F),D_{\left(p,\theta,q,\nu\right)}^{L}\right)$
is a normed space.

###### Proof.

We prove the triangle inequality. Let $i=1,2$ and
$T_{i}\in\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$. For each
$\epsilon>0$, there exists a Banach spaces $G_{i}$ and $H_{i}$, a Lipschitz
operators $S_{i}\in\Pi_{p}^{L}(X,G_{i})$, a bounded operators
$R_{i}\in\Pi_{q}(F^{*},H_{i})$ and a positive constants $C_{i}$ such that

$\left|\left\langle
T_{i}x^{\prime}-T_{i}x^{\prime\prime},b^{*}\right\rangle\right|\leq
C_{i}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|S_{i}x^{\prime}-S_{i}x^{\prime\prime}|G_{i}\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|R_{i}(b^{*})|H_{i}\right\|^{1-\nu}\forall
x^{\prime},x^{\prime\prime}\in X\;\forall b^{*}\subset F^{*}$ (19)

and

$C_{i}\cdot\pi_{p}^{L}(S_{i})^{1-\theta}\cdot\pi_{q}(R_{i})^{1-\nu}\leq
D_{\left(p,\theta,q,\nu\right)}^{L}(T_{i})+\epsilon.$ (20)

For $x^{\prime},x^{\prime\prime}\in X$ and $b^{*}$ we have

$\displaystyle\left|\left\langle
T_{i}x^{\prime}-T_{i}x^{\prime\prime},b^{*}\right\rangle\right|$
$\displaystyle\leq C_{i}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|S_{i}x^{\prime}-S_{i}x^{\prime\prime}|G_{i}\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|R_{i}(b^{*})|H_{i}\right\|^{1-\nu}$
$\displaystyle=\widetilde{C}_{i}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|\widetilde{S}_{i}x^{\prime}-\tilde{S}_{i}x^{\prime\prime}|G_{i}\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|\widetilde{R}_{i}(b^{*})|H_{i}\right\|^{1-\nu},$

where
$\widetilde{C}_{i}=C_{i}^{\frac{1}{r}}\cdot\pi_{p}^{L}(S_{i})^{\frac{1-\theta}{r}}\cdot\pi_{q}(R_{i})^{\frac{1-\nu}{r}}$,
$\widetilde{S}_{i}=C_{i}^{\frac{1}{p}}\cdot\pi_{p}^{L}(S_{i})^{\frac{1-\theta}{p}}\cdot\pi_{q}(R_{i})^{\frac{1-\nu}{p}}\frac{S_{i}}{\pi_{p}^{L}(S_{i})}$,
and
$\widetilde{R}_{i}=C_{i}^{\frac{1}{q}}\cdot\pi_{q}^{L}(S_{i})^{\frac{1-\theta}{q}}\cdot\pi_{q}(R_{i})^{\frac{1-\nu}{p}}\frac{R_{i}}{\pi_{q}^{L}(R_{i})}$.

From (19) and (20) we have

$C_{i}\leq\left(D_{\left(p,\theta,q,\nu\right)}^{L}(T_{i})+\epsilon\right)^{\frac{1}{r}}.$

$\pi_{p}^{L}(S_{i})\leq\left(D_{\left(p,\theta,q,\nu\right)}^{L}(T_{i})+\epsilon\right)^{\frac{1}{p}}\
\text{and}\
\pi_{q}(R_{i})\leq\left(D_{\left(p,\theta,q,\nu\right)}^{L}(T_{i})+\epsilon\right)^{\frac{1}{q}}.$
(21)

Let $G$ and $H$ be a Banach spaces obtained as a direct sum of $\ell_{p}$ and
$\ell_{q}$ by $G_{1}$ and $G_{2}$ and $H_{1}$ and $H_{2}$ respectively. Let
$S$ be a Lipschitz operator from $X$ into $G$ such that
$S(x)=(S_{i}(x))_{i=1}^{2}$ for $x\in X$ and $R$ be a bounded operator from
$F^{*}$ into $H$ such that $R(b)=(R_{i}(b))_{i=1}^{2}$ for $b\in F^{*}$. For
each finite sequence $x^{\prime}$, $x^{\prime\prime}$ in $X$ we have

$\displaystyle\left\|(S(x^{\prime}_{j})-S(x^{\prime\prime}_{j}))_{j=1}^{n}|\ell_{p}(G)\right\|$
$\displaystyle=\left[\sum\limits_{j=1}^{n}\left\|S(x^{\prime}_{j})-S(x^{\prime\prime}_{j})|G\right\|^{p}\right]^{\frac{1}{p}}=\left[\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{2}\left\|S_{i}(x^{\prime}_{j})-S_{i}(x^{\prime\prime}_{j})|G\right\|^{p}\right]^{\frac{1}{p}}$
$\displaystyle\leq\left[\sum\limits_{i=1}^{2}\pi_{p}^{L}(S_{i})^{p}\sup\limits_{f\in
B_{{X}^{\\#}}}\sum\limits_{j=1}^{n}\left|fx^{\prime}_{j}-fx^{\prime\prime}_{j}\right|^{p}\right]^{\frac{1}{p}}$
$\displaystyle=\sup\limits_{f\in
B_{{X}^{\\#}}}\left[\sum\limits_{j=1}^{n}\left|fx^{\prime}_{j}-fx^{\prime\prime}_{j}\right|^{p}\right]^{\frac{1}{p}}\left(\sum\limits_{i=1}^{2}\pi_{p}^{L}(S_{i})^{p}\right)^{\frac{1}{p}}$

$\pi_{p}^{L}(S)\leq\left(\sum\limits_{i=1}^{2}\pi_{p}^{L}(S_{i})^{p}\right)^{\frac{1}{p}}\leq\left(D_{\left(p,\theta,q,\nu\right)}^{L}(T_{1})+D_{\left(p,\theta,q,\nu\right)}^{L}(T_{2})+2\epsilon\right)^{\frac{1}{p}}.$
(22)

$\pi_{q}(B)\leq\left(D_{\left(p,\theta,q,\nu\right)}^{L}(T_{1})+D_{\left(p,\theta,q,\nu\right)}^{L}(T_{2})+2\epsilon\right)^{\frac{1}{q}}.$
(23)

$\displaystyle\left|\left\langle(T_{1}+T_{1})x^{\prime}-(T_{1}+T_{1})x^{\prime\prime},b^{*}\right\rangle\right|\leq\sum\limits_{i=1}^{2}C_{i}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|S_{i}x^{\prime}-S_{i}x^{\prime\prime}|G_{i}\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|R_{i}(b^{*})|H_{i}\right\|^{1-\nu}$
$\displaystyle\leq
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|b^{*}\right\|^{\nu}\left(\sum\limits_{i=1}^{2}C_{i}^{r}\right)^{\frac{1}{r}}\left(\sum\limits_{i=1}^{2}\left\|S_{i}x^{\prime}-S_{i}x^{\prime\prime}|G_{i}\right\|^{p}\right)^{\frac{1-\theta}{p}}\left(\sum\limits_{i=1}^{2}\left\|R_{i}(b^{*})|H_{i}\right\|^{q}\right)^{\frac{1-\nu}{q}}$
$\displaystyle=d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|b^{*}\right\|^{\nu}\left(\sum\limits_{i=1}^{2}C_{i}^{r}\right)^{\frac{1}{r}}\left\|Sx^{\prime}-Sx^{\prime\prime}|G\right\|^{1-\theta}\left\|R(b^{*})|H\right\|^{1-\nu}$

$\displaystyle D_{\left(p,\theta,q,\nu\right)}^{L}(T_{1}+T_{2})$
$\displaystyle\leq\left(\sum\limits_{i=1}^{2}C_{i}^{r}\right)^{\frac{1}{r}}\pi_{p}^{L}(S)^{1-\theta}\pi_{q}(R)^{1-\nu}$
$\displaystyle\leq\left(D_{\left(p,\theta,q,\nu\right)}^{L}(T_{1})+D_{\left(p,\theta,q,\nu\right)}^{L}(T_{2})+2\epsilon\right)^{\frac{1}{r}+\frac{1-\theta}{p}+\frac{1-\nu}{q}}.$

Hence $D_{\left(p,\theta,q,\nu\right)}^{L}(T_{1}+T_{2})\leq
D_{\left(p,\theta,q,\nu\right)}^{L}(T_{1})+D_{\left(p,\theta,q,\nu\right)}^{L}(T_{2})$

∎

######  Remark 6.

If $\theta=\nu=0$, then the class of all Lipschitz
$\left(p,\theta,q,\nu\right)$-dominated operators from $X$ to $F$ are the
class of all Lipschitz $\left(p,q\right)$-dominated operators from $X$ to $F$
considered in [6] with
$\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)=\mathcal{D}_{\left(p,q\right)}^{L}(X,F)$.

######  Theorem 3.

Let $X$ be a metric space, $F$ be a Banach space and $T\in Lip(X,F)$. The
following conditions are equivalent.

  1. $\bf(1)$

$T\in\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$.

  2. $\bf(2)$

There is a constant $C\geq 0$ and regular probabilities $\mu$ and $\tau$ on
$B_{X^{\\#}}$ and $B_{F^{**}}$, respectively such that for every $x^{\prime}$,
$x^{\prime\prime}$ in $X$ and $b^{*}$ in $F^{*}$ the following inequality
holds

$\displaystyle\left|\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|\leq C$
$\displaystyle\cdot\left[\int\limits_{B_{X^{\\#}}}\left(\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{1-\theta}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\right)^{\frac{p}{1-\theta}}d\mu(f)\right]^{\frac{1-\theta}{p}}$
$\displaystyle\cdot\left[\int\limits_{B_{F^{**}}}\left(\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{1-\nu}\left\|b^{*}\right\|^{\nu}\right)^{\frac{q}{1-\nu}}d\tau(b^{**})\right]^{\frac{1-\nu}{q}}.$

  3. $\bf(3)$

There exists a constant $C\geq 0$ such that for every finite sequences
$x^{\prime}$, $x^{\prime\prime}$ in $X$; $\sigma$ in $\mathbb{R}$ and
$y^{*}\subset F^{*}$ the inequality

$\left\|\sigma\cdot\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle|\ell_{r^{\prime}}\right\|\leq
C\cdot\left\|(\sigma,x^{\prime},x^{\prime\prime})\Big{|}\delta_{p,\theta}^{L}(\mathbb{R}\times
X\times X)\right\|\left\|b^{*}\Big{|}\delta_{q,\nu}(F^{*})\right\|$ (24)

holds.

In this case, $D_{\left(p,\theta,q,\nu\right)}^{L}$ is equal to the infimum of
such constants $C$ in either $\bf(2)$, or $\bf(3)$.

###### Proof.

$\bf(1)\Longrightarrow\bf(2)$ If
$T\in\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$, then there exists a
Banach spaces $G$ and $H$, a Lipschitz operator $S_{1}\in\Pi_{p}^{L}(X,G)$, a
bounded operator $S_{2}\in\Pi_{q}(F^{*},H)$ and a positive constant $C$ such
that

$\left|\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|\leq C\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|S_{1}x^{\prime}-S_{1}x^{\prime\prime}|G\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|S_{2}(b^{*})|H\right\|^{1-\nu}$
(25)

for arbitrary finite sequences $x^{\prime}$, $x^{\prime\prime}$ in $X$ and
$b^{*}\subset F^{*}$. Since $S_{1}$ is Lipschitz $p$–summing operator and
$S_{2}$ is $q$–summing operator then there exists regular probabilities $\mu$
and $\tau$ on $B_{X^{\\#}}$ and $B_{F^{**}}$, respectively such that

$\displaystyle\left|\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|\leq
C\cdot\pi_{p}^{L}(S_{1})^{1-\theta}\cdot\pi_{q}(S_{2})^{1-\nu}$
$\displaystyle\left[\int\limits_{B_{X^{\\#}}}\left(\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{1-\theta}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\right)^{\frac{p}{1-\theta}}d\mu(f)\right]^{\frac{1-\theta}{p}}$
$\displaystyle\cdot\left[\int\limits_{B_{F^{**}}}\left(\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{1-\nu}\left\|b^{*}\right\|^{\nu}\right)^{\frac{q}{1-\nu}}d\tau(b^{**})\right]^{\frac{1-\nu}{q}}.$

$\bf(2)\Longrightarrow\bf(1)$ Let $x^{\prime}$, $x^{\prime\prime}$ be finite
sequences in $X$ and $y^{*}$ be a finite sequence in $F^{*}$. Let
$\varphi_{b^{*}}:=\left\langle b^{*},b^{**}\right\rangle$. For each $x\in X$,
let $\delta_{(x,0)}:X^{\\#}\longrightarrow\mathbb{R}$ be the linear map
defined by

$\delta_{(x,0)}(f)=f(x)\ \ \ (f\in X^{\\#}).$

By setting $S_{1}x:=\delta_{(x,0)}$, $S_{2}b^{*}:=\varphi_{b^{*}}$,
$G:=L_{p}(B_{X^{\\#}},\mu)$, and $H:=L_{q}(B_{F^{**}},\tau)$ we obtain a
Lipschitz operator $S_{1}\in\Pi_{p}^{L}(X,G)$ with $\pi_{p}^{L}(S_{1})\leq 1$
and an operator $S_{2}\in\Pi_{q}(F^{*},H)$ with $\pi_{q}(S_{2})\leq 1$ such
that

$\displaystyle\left|\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|$ $\displaystyle\leq
C\cdot\left[\int\limits_{B_{X^{\\#}}}\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{p}d\mu(f)\right]^{\frac{1-\theta}{p}}\left[\int\limits_{B_{F{**}}}\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{q}d\tau(b^{**})\right]^{\frac{1-\nu}{q}}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|b^{*}\right\|^{\nu}$
$\displaystyle=C\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\left\|S_{1}x^{\prime}-S_{1}x^{\prime\prime}|G\right\|^{1-\theta}\left\|b^{*}\right\|^{\nu}\left\|S_{2}(b^{*})|H\right\|^{1-\nu}$

$\bf(2)\Longrightarrow\bf(3)$ Let $x^{\prime}$, $x^{\prime\prime}$ be finite
sequences in $X$; $\sigma$ in $\mathbb{R}$ and $y^{*}$ be a finite sequence in
$F^{*}$. By $\bf(2)$ and using the Hölder inequality with exponents of
$1=\frac{1}{r}+\frac{1-\theta}{p}+\frac{1-\nu}{q}$ we have

$\displaystyle\left[\sum\limits_{j=1}^{m}\left|\sigma_{j}\right|^{r^{\prime}}\left|\left\langle
Tx^{\prime}_{j}-Tx^{\prime\prime}_{j},b_{j}^{*}\right\rangle\right|^{r^{\prime}}\right]^{\frac{1}{r^{\prime}}}\leq
C$
$\displaystyle\cdot\left[\sum\limits_{j=1}^{m}\left(\int\limits_{B_{X^{\\#}}}\left(\left|\sigma_{j}\right|\left|f(x^{\prime}_{j})-f(x^{\prime\prime}_{j})\right|^{1-\theta}d_{X}(x^{\prime}_{j},x^{\prime\prime}_{j})^{\theta}\right)^{\frac{p}{1-\theta}}d\mu(f)\right)\right]^{\frac{1-\theta}{p}}$
$\displaystyle\cdot\left[\sum\limits_{j=1}^{m}\left(\int\limits_{B_{F^{**}}}\left(\left|\left\langle
b_{j}^{*},b^{**}\right\rangle\right|^{1-\nu}\left\|b_{j}^{*}\right\|^{\nu}\right)^{\frac{q}{1-\nu}}d\tau(b^{**})\right)\right]^{\frac{1-\nu}{q}}$
$\displaystyle\leq C\cdot\sup\limits_{f\in
B_{{X}^{\\#}}}\Bigg{[}\sum\limits_{j=1}^{m}\left[\left|\sigma_{j}\right|\left|fx^{\prime}_{j}-fx^{\prime\prime}_{j}\right|^{1-\theta}d_{X}(x^{\prime}_{j},x^{\prime\prime}_{j})^{\theta}\right]^{\frac{p}{1-\theta}}\Bigg{]}^{\frac{1-\theta}{p}}$
$\displaystyle\cdot\sup\limits_{b^{**}\in
B_{{F}^{**}}}\Bigg{[}\sum\limits_{j=1}^{m}\left[\left|\left\langle
b^{*}_{j},b^{**}\right\rangle\right|^{1-\nu}\left\|b^{*}_{j}\right\|^{\nu}\right]^{\frac{q}{1-\nu}}\Bigg{]}^{\frac{1-\nu}{q}}$
$\displaystyle=C\cdot\left\|(\sigma,x^{\prime},x^{\prime\prime})\Big{|}\delta_{p,\theta}^{L}(\mathbb{R}\times
X\times X)\right\|\left\|b^{*}\Big{|}\delta_{q,\nu}(F^{*})\right\|$

$\bf(3)\Longrightarrow\bf(2)$ Take $\left[C(B_{X^{\\#}})\times
C(B_{F^{**}})\right]^{*}$ equipped with the weak $C(B_{X^{\\#}})\times
C(B_{F^{**}})$-topology. Then $W(B_{X^{\\#}})\times W(B_{F^{**}})$ is a
compact convex subset. For any finite sequences $\sigma$ in $\mathbb{R}$,
$x^{\prime}$, $x^{\prime\prime}$ in $X$ and $y^{*}$ in $F^{*}$ the equation

$\displaystyle\Psi(\mu,\tau)=\sum\limits_{j=1}^{n}\bigg{(}\frac{1}{r^{\prime}}\left|\sigma_{j}\left\langle
Tx^{\prime}_{j}-Tx^{\prime\prime}_{j},b^{*}_{j}\right\rangle\right|^{r^{\prime}}$
$\displaystyle-\frac{C^{r^{\prime}}}{\frac{p}{1-\theta}}\int\limits_{B_{X^{\\#}}}\left|\sigma_{j}\right|^{\frac{p}{1-\theta}}d_{X}(x^{\prime}_{j},x^{\prime\prime}_{j})^{\frac{\theta
p}{1-\theta}}\left|f(x^{\prime}_{j})-f(x^{\prime\prime}_{j})\right|^{p}d\mu(f)$
$\displaystyle-\frac{C^{r^{\prime}}}{\frac{q}{1-\nu}}\int\limits_{B_{F^{**}}}\left\|b^{*}_{j}\right\|^{\frac{\nu
q}{1-\nu}}\left|\left\langle
b_{j}^{*},b^{**}\right\rangle\right|^{q}d\tau(b^{**})\bigg{)}$

defines a continuous convex function $\Psi$ on $W(B_{X^{\\#}})\times
W(B_{F^{**}})$. From the compactness of $B_{X^{\\#}}$ and $B_{F^{**}}$, there
exists $f_{0}\in B_{X^{\\#}}$ and $y^{**}_{0}\in B_{F^{**}}$ such that

$\zeta=\Bigg{[}\sum\limits_{j=1}^{n}\left[\left|\sigma_{j}\right|\left|f_{0}\>x^{\prime}_{j}-f_{0}\>x^{\prime\prime}_{j}\right|^{1-\theta}d_{X}(x^{\prime}_{j},x^{\prime\prime}_{j})^{\theta}\right]^{\frac{p}{1-\theta}}\Bigg{]}^{\frac{1-\theta}{p}}.$

and

$\beta=\Bigg{[}\sum\limits_{j=1}^{n}\left[\left|\left\langle
b^{*}_{j},b^{**}_{0}\right\rangle\right|^{1-\nu}\left\|b^{*}_{j}\right\|^{\nu}\right]^{\frac{q}{1-\nu}}\Bigg{]}^{\frac{1-\nu}{q}}.$

If $\delta(f_{0})$ and $\delta(y^{**}_{0})$ denotes the Dirac measure at the
point $f_{0}$ and in $y^{**}_{0}$, respectively, then we have

$\displaystyle\Psi\left(\delta(f_{0}),\delta(b^{**}_{0})\right)=\frac{1}{r^{\prime}}\sum\limits_{j=1}^{n}\bigg{(}\left|\sigma_{j}\left\langle
Tx^{\prime}_{j}-Tx^{\prime\prime}_{j},b^{*}_{j}\right\rangle\right|^{r^{\prime}}$
$\displaystyle-\frac{C^{r^{\prime}}}{\frac{p}{1-\theta}}\left|\sigma_{j}\right|^{\frac{p}{1-\theta}}d_{X}(x^{\prime}_{j},x^{\prime\prime}_{j})^{\frac{\theta
p}{1-\theta}}\left|f_{0}(x^{\prime}_{j})-f_{0}(x^{\prime\prime}_{j})\right|^{p}$
$\displaystyle-\frac{C^{r^{\prime}}}{\frac{q}{1-\nu}}\left\|b^{*}_{j}\right\|^{\frac{\nu
q}{1-\nu}}\left|\left\langle
b_{j}^{*},b_{0}^{**}\right\rangle\right|^{q}\bigg{)}$
$\displaystyle=\frac{1}{r^{\prime}}\sum\limits_{j=1}^{n}\left|\sigma_{j}\right|^{r^{\prime}}\left|\left\langle
Tx^{\prime}_{j}-Tx^{\prime\prime}_{j},b^{*}_{j}\right\rangle\right|^{r^{\prime}}-C^{r^{\prime}}\left(\frac{1-\theta}{p}\zeta^{\frac{p}{1-\theta}}+\frac{1-\nu}{q}\beta^{\frac{p}{1-\theta}}\right)$
$\displaystyle\leq\frac{1}{r^{\prime}}\left(\sum\limits_{j=1}^{n}\left|\sigma_{j}\right|^{r^{\prime}}\left|\left\langle
Tx^{\prime}_{j}-Tx^{\prime\prime}_{j},b^{*}_{j}\right\rangle\right|^{r^{\prime}}\right)^{r^{\prime}}-\frac{C^{r^{\prime}}}{r^{\prime}}(\zeta\cdot\beta)^{r^{\prime}}$
$\displaystyle=\frac{1}{r^{\prime}}\left[\left(\sum\limits_{j=1}^{n}\left|\sigma_{j}\right|^{r^{\prime}}\left|\left\langle
Tx^{\prime}_{j}-Tx^{\prime\prime}_{j},b^{*}_{j}\right\rangle\right|^{r^{\prime}}\right)^{r^{\prime}}-(C\cdot\zeta\cdot\beta)^{r^{\prime}}\right]$
$\displaystyle\leq 0.$

Since the collection $\mathfrak{Q}$ of all functions $\Psi$ obtained in this
way is concave, by [14, E.4.2] there are $\mu_{0}\in W(B_{X^{\\#}})$ and
$\tau_{0}\in W(B_{F^{**}})$ such that $\Psi\left(\mu_{0},\tau_{0}\right)\leq
0$ for all $\Psi\in\mathfrak{Q}$. In particular, if $\Psi$ is generated by
single finite sequences $\sigma$ in $\mathbb{R}$, $x^{\prime}$,
$x^{\prime\prime}$ in $X$ and $y^{*}$ in $F^{*}$, it follows that

$\displaystyle\frac{1}{r^{\prime}}\left|\sigma\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|^{r^{\prime}}$
$\displaystyle-\frac{C^{r^{\prime}}}{\frac{p}{1-\theta}}\int\limits_{B_{X^{\\#}}}\left|\sigma\right|^{\frac{p}{1-\theta}}d_{X}(x^{\prime},x^{\prime\prime})^{\frac{\theta
p}{1-\theta}}\left|fx^{\prime}-fx^{\prime\prime}\right|^{p}d\mu_{0}(f)$
$\displaystyle-\frac{C^{r^{\prime}}}{\frac{q}{1-\nu}}\int\limits_{B_{F^{**}}}\left\|b^{*}\right\|^{\frac{\nu
q}{1-\nu}}\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{q}d\tau_{0}(b^{**})\leq 0.$

Finally, we put

$s_{1}:=\left[\int\limits_{B_{X^{\\#}}}\left|\sigma\right|^{\frac{p}{1-\theta}}d_{X}(x^{\prime},x^{\prime\prime})^{\frac{\theta
p}{1-\theta}}\left|fx^{\prime}-fx^{\prime\prime}\right|^{p}d\mu_{0}(f)\right]^{\frac{1-\theta}{p}}$

and

$s_{2}:=\left[\int\limits_{B_{F^{**}}}\left\|b^{*}\right\|^{\frac{\nu
q}{1-\nu}}\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{q}d\tau_{0}(b^{**})\right]^{\frac{1-\nu}{q}}.$

Then

$\displaystyle\left|\sigma\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|$
$\displaystyle=s_{1}s_{2}\left|(s_{1}^{-1}\sigma)\left\langle
Tx^{\prime}-Tx^{\prime\prime},s_{2}^{-1}b^{*}\right\rangle\right|$
$\displaystyle\leq
Cs_{1}s_{2}\bigg{[}\frac{r^{\prime}}{\frac{p}{1-\theta}}\int\limits_{B_{X^{\\#}}}\left|s_{1}^{-1}\sigma\right|^{\frac{p}{1-\theta}}d_{X}(x^{\prime},x^{\prime\prime})^{\frac{\theta
p}{1-\theta}}\left|fx^{\prime}-fx^{\prime\prime}\right|^{p}d\mu_{0}(f)$
$\displaystyle+\frac{r^{\prime}}{\frac{q}{1-\nu}}\int\limits_{B_{F^{**}}}\left\|s_{2}^{-1}b^{*}\right\|^{\frac{\nu
q}{1-\nu}}\left|\left\langle
s_{2}^{-1}b^{*},b^{**}\right\rangle\right|^{q}d\tau_{0}(b^{**})\bigg{]}^{\frac{1}{r^{\prime}}}.$
$\displaystyle\leq C\ s_{1}\ s_{2}.$

Hence

$\displaystyle\left|\left\langle
Tx^{\prime}-Tx^{\prime\prime},b^{*}\right\rangle\right|\leq C$
$\displaystyle\cdot\left[\int\limits_{B_{X^{\\#}}}\left(\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{1-\theta}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\right)^{\frac{p}{1-\theta}}d\mu(f)\right]^{\frac{1-\theta}{p}}$
$\displaystyle\cdot\left[\int\limits_{B_{F^{**}}}\left(\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{1-\nu}\left\|b^{*}\right\|^{\nu}\right)^{\frac{q}{1-\nu}}d\tau(b^{**})\right]^{\frac{1-\nu}{q}}.$

∎

The aforementioned Theorem 3 will use to prove the next result.

######  Corollary 1.

The linear space $\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$ is a
Banach space under the norm $D_{\left(p,\theta,q,\nu\right)}^{L}$.

###### Proof.

To prove that $\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$ is complete
space. We consider an arbitrary Cauchy sequence
$\left(T_{n}\right)_{n\in\mathbb{N}}$ in
$\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$ and show that
$\left(T_{n}\right)_{n\in\mathbb{N}}$ converges to
$T\in\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$. Since
$\left(T_{n}\right)_{n\in\mathbb{N}}$ is Cauchy, for every $\epsilon>0$ there
is an $n_{0}$ such that

$D_{\left(p,\theta,q,\nu\right)}^{L}\left(T_{m}-T_{n}\right)\leq\epsilon\ \
\text{for}\ \ m,n\geq n_{0},$ (26)

Since $\operatorname{Lip}\left(T_{m}-T_{n}\right)\leq
D_{\left(p,\theta,q,\nu\right)}^{L}\left(T_{m}-T_{n}\right)$ then
$\left(T_{n}\right)_{n\in\mathbb{N}}$ is also a Cauchy sequence in the Banach
space $\operatorname{Lip}(X,F)$, and there is a Lipschitz map $T$ with

$\lim_{n\rightarrow\infty}\operatorname{Lip}\left(T-T_{n}\right)=0.$

From $\bf(2)$ of Theorem 3 given $\epsilon>0$ there is $n_{0}\in\mathbb{N}$
such that, for each $n,m\in\mathbb{N}$, $n,m\geq n_{0}$, there exist
probabilities $\mu_{nm}$ on $B_{X^{\\#}}$ and $\tau_{nm}$ on $B_{F^{**}}$ such
that for every $x^{\prime}$, $x^{\prime\prime}$ in $X$ and $b^{*}$ in $F^{*}$
the following inequality holds

$\displaystyle\left|\left\langle(T_{m}-T_{n})\;x^{\prime}-(T_{m}-T_{n})\;x^{\prime\prime},b^{*}\right\rangle\right|\leq\epsilon\;d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\;\left\|b^{*}\right\|^{\nu}$
$\displaystyle\cdot\left[\int\limits_{B_{X^{\\#}}}\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{p}d\mu_{nm}(f)\right]^{1-\theta}$
$\displaystyle\cdot\left[\int\limits_{B_{F^{**}}}\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{q}d\tau_{nm}(b^{**})\right]^{1-\nu}.$

Fixed $n\geq n_{0}$, by the weak compactness of $W\left(B_{X^{\\#}}\right)$
and $W\left(B_{F^{**}}\right)$, there is a sub-net
$\left(\mu_{nm}(\alpha),\tau_{nm}(\alpha)\right)_{\alpha\in\mathcal{A}}$
convergent to $\left(\mu_{n},\tau_{n}\right)\in W(B_{X^{\\#}})\times
W(B_{F^{**}})$ in the topology $\sigma\left((C(B_{X^{\\#}}\times
C(B_{F^{**}})))^{\ast},C(B_{X^{\\#}})\times C(B_{F^{**}})\right)$. Then, there
is $\alpha_{0}\in\mathcal{A}$ such that for each $x^{\prime}$,
$x^{\prime\prime}$ in $X$, $b^{*}$ in $F^{*}$ and $\alpha\in\mathcal{A}$ with
$\alpha\geq\alpha_{0}$ we have

$\displaystyle\left|\left\langle(T_{m(\alpha)}-T_{n})\;x^{\prime}-(T_{m(\alpha)}-T_{n})\;x^{\prime\prime},b^{*}\right\rangle\right|\leq\epsilon\;d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\;\left\|b^{*}\right\|^{\nu}$
$\displaystyle\cdot\left[\int\limits_{B_{X^{\\#}}}\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{p}d(\mu_{{nm}(\alpha)}-\mu_{n})(f)+\int\limits_{B_{X^{\\#}}}\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{p}d\mu_{n}(f)\right]^{1-\theta}$
$\displaystyle\cdot\left[\int\limits_{B_{F^{**}}}\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{q}d(\tau_{{nm}(\alpha)}-\tau_{n})(b^{**})+\int\limits_{B_{F^{**}}}\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{q}d\tau_{n}(b^{**})\right]^{1-\nu}.$

and taking limits when $\alpha\in\mathcal{A}$ we have

$\displaystyle\left|\left\langle(T-T_{n})\;x^{\prime}-(T-T_{n})\;x^{\prime\prime},b^{*}\right\rangle\right|\leq\epsilon\;d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\;\left\|b^{*}\right\|^{\nu}$
$\displaystyle\cdot\left[\int\limits_{B_{X^{\\#}}}\left|f(x^{\prime})-f(x^{\prime\prime})\right|^{p}d\mu_{n}(f)\right]^{1-\theta}$
$\displaystyle\cdot\left[\int\limits_{B_{F^{**}}}\left|\left\langle
b^{*},b^{**}\right\rangle\right|^{q}d\tau_{n}(b^{**})\right]^{1-\nu}$

for every $x^{\prime}$, $x^{\prime\prime}$ in $X$ and $b^{*}$ in $F^{*}$. It
follows that $T-T_{n}\in\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$
and therefore $T\in\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$. From
the last inequality it follows that
$D_{\left(p,\theta,q,\nu\right)}^{L}(T-T_{n})\leq\epsilon$ if $n\geq n_{0}$
and hence $\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L}(X,F)$ is a Banach
space. ∎

By Proposition 22, Theorem 3, and Corollary 1 we obtain the following result.

######  Proposition 23.

$\left[\mathcal{D}_{\left(p,\theta,q,\nu\right)}^{L},D_{\left(p,\theta,q,\nu\right)}^{L}\right]$
is a Banach nonlinear ideal.

######  Remark 7.

Definition 3 can be generalized as follows. Let $0\leq\theta<1$ and
$\left[\textfrak{A}^{L},\mathbf{A}^{L}\right]$ be a normed nonlinear ideal. A
Lipschitz operator $T$ from $X$ into $F$ belongs to
$\left(\textfrak{A}^{L},\textfrak{B}^{L}\right)_{\theta}(X,F)$ if there exist
a Banach spaces $G_{1}$, $G_{2}$ and a Lipschitz operators
$S_{1}\in\textfrak{A}^{L}(X,G_{1})$ and $S_{2}\in\textfrak{B}^{L}(X,G_{2})$
such that

$\left\|Tx^{\prime}-Tx^{\prime\prime}|F\right\|\leq\left\|S_{1}x^{\prime}-S_{1}x^{\prime\prime}|G_{1}\right\|^{1-\theta}\cdot\left\|S_{2}x^{\prime}-S_{2}x^{\prime\prime}|G_{2}\right\|^{\theta},\
\ \forall\>x^{\prime},\;x^{\prime\prime}\in X.$ (27)

For each $T\in\left(\textfrak{A}^{L},\textfrak{B}^{L}\right)_{\theta}(X,F)$,
we set

$\left(\mathbf{A}^{L},\mathbf{B}^{L}\right)_{\theta}(T):=\inf\mathbf{A}^{L}(S_{1})^{1-\theta}\cdot\mathbf{B}^{L}(S_{2})^{\theta}$
(28)

where the infimum is taken over all Lipschitz operators $S_{1}$, $S_{2}$
admitted in (27).

Note that
$\operatorname{Lip}(T)\leq\left(\mathbf{A}^{L},\mathbf{B}^{L}\right)_{\theta}(T)$,
by definition. The nonlinear ideal
$\left[\textfrak{A}^{L}_{\theta},\mathbf{A}^{L}_{\theta}\right]$ now appear as
$\left[\left(\textfrak{A}^{L},\operatorname{Lip}\right)_{\theta},\left(\mathbf{A}^{L},\operatorname{Lip}(\cdot)\right)_{\theta}\right]$.

##  4 Nonlinear operator ideals between metric spaces

######  Definition 5.

Suppose that, for every pair of metric spaces $X$ and $Y$, we are given a
subset $\mathscr{A}^{L}(X,Y)$ of $\mathscr{L}(X,Y)$. The class

$\mathscr{A}^{L}:=\bigcup_{X,Y}\mathscr{A}^{L}(X,Y)$

is said to be a nonlinear operator ideal, or just a nonlinear ideal, if the
following conditions are satisfied:

  1. $\bf(\widetilde{\widetilde{NOI_{0}\,}})$

If $Y=F$ is a Banach space, then $g\boxdot e\in\mathscr{A}^{L}(X,F)$ for $g\in
X^{\\#}$ and $e\in F$.

  2. $\bf(\widetilde{\widetilde{NOI_{1}\,}})$

$BTA\in\mathscr{A}^{L}(X_{0},Y_{0})$ for $A\in\mathscr{L}(X_{0},X)$,
$T\in\mathscr{A}^{L}(X,Y)$, and $B\in\mathscr{L}(Y,Y_{0})$.

Condition $\bf(\widetilde{\widetilde{NOI_{0}}})$ implies that
$\mathscr{A}^{L}$ contains nonzero Lipschitz operators.

###  4.1 Lipschitz interpolative ideal procedure between metric spaces

######  Definition 6.

Let $0\leq\theta<1$. A Lipschitz map $T$ from $X$ into $Y$ belongs to
$\mathscr{A}^{L}_{\theta}(X,Y)$ if there exist a constant $C\geq 0$, a metric
space $Z$ and a Lipschitz map $S\in\mathscr{A}^{L}(X,Z)$ such that

$d_{Y}(Tx^{\prime},Tx^{\prime\prime})\leq C\cdot
d_{Z}(Sx^{\prime},Sx^{\prime\prime})^{1-\theta}\cdot
d_{X}(x^{\prime},x^{\prime\prime})^{\theta},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X.\neq$ (29)

For each $T\in\mathscr{A}^{L}_{\theta}(X,Y)$, we set

$\mathbf{A}^{L}_{\theta}(T):=\inf C\mathbf{A}^{L}(S)^{1-\theta}$ (30)

where the infimum is taken over all Lipschitz operators $S$ admitted in (29).

######  Proposition 24.

$\mathscr{A}^{L}_{\theta}$ is a nonlinear ideal with
$\mathbf{A}^{L}_{\theta}(BTA)\leq\operatorname{Lip}(B)\cdot\mathbf{A}^{L}_{\theta}(T)\cdot\operatorname{Lip}(A)$
for $A\in\mathscr{L}(X_{0},X)$, $T\in\mathscr{A}^{L}(X,Y)$, and
$B\in\mathscr{L}(Y,Y_{0})$..

###### Proof.

The proof of condition $\bf(\widetilde{\widetilde{NOI_{0}\,}})$ is similar to
the proof of algebraic condition $\bf(\widetilde{PNOI_{0}})$ in Proposition
20. To prove the condition $\bf(\widetilde{\widetilde{NOI_{1}\,}})$, let
$A\in\mathscr{L}(X_{0},X)$, $T\in\mathscr{A}^{L}_{\theta}(X,Y)$, and
$B\in\mathscr{L}(Y,Y_{0})$ and $x^{\prime}_{0},x^{\prime\prime}_{0}$ in $X$.
Then

$\displaystyle d_{Y_{0}}(BTAx^{\prime}_{0},BTAx^{\prime\prime}_{0})$
$\displaystyle\leq\operatorname{Lip}(B)\cdot
d_{Y}(TAx^{\prime}_{0},TAx^{\prime\prime}_{0})$
$\displaystyle\leq\operatorname{Lip}(B)\cdot C\cdot d_{Z}(S\circ
A(x^{\prime}),S\circ A(x^{\prime\prime}))^{1-\theta}\cdot
d_{X}(Ax^{\prime},Ax^{\prime\prime})^{\theta}$ $\displaystyle\leq
C\cdot\operatorname{Lip}(B)\cdot\operatorname{Lip}(A)^{\theta}\cdot
d_{Z}(S\circ A(x^{\prime}),S\circ
A(x^{\prime\prime}))^{1-\theta}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}.$

Since a Lipschitz map $\widetilde{S}:=S\circ A\in\mathscr{A}^{L}(X_{0},Z)$ and
$\widetilde{C}:=C\cdot\operatorname{Lip}(B)\cdot\operatorname{Lip}(A)^{\theta}$,
hence $BTA\in\mathscr{A}^{L}_{\theta}(X_{0},Y_{0})$. Moreover, from (30) we
have

$\displaystyle\mathbf{A}^{L}_{\theta}(BTA)$
$\displaystyle:=\inf\widetilde{C}\cdot\mathbf{A}^{L}(\widetilde{S})^{1-\theta}$
$\displaystyle\leq
C\cdot\operatorname{Lip}(B)\cdot\operatorname{Lip}(A)^{\theta}\cdot\mathbf{A}^{L}(S\circ
A)^{1-\theta}$
$\displaystyle=C\cdot\operatorname{Lip}(B)\cdot\operatorname{Lip}(A)\cdot\mathbf{A}^{L}(S)^{1-\theta}.$
(31)

Taking the infimum over all such $S\in\mathscr{A}^{L}(X,Z)$ on the right side
of (4.1), we have

$\mathbf{A}^{L}_{\theta}(BTA)\leq\operatorname{Lip}(B)\cdot\mathbf{A}^{L}_{\theta}(T)\cdot\operatorname{Lip}(A).$

∎

###  4.2 Basic Examples of Lipschitz interpolative ideal procedure

  1. $\bf(1)$

Lipschitz $\left(p,s,\theta\right)$-summing maps

A Lipschitz map $T$ from $X$ to $Y$ is called Lipschitz
$\left(p,s,\theta\right)$-summing if there is a constant $C\geq 0$ such that

$\left\|(\sigma,Tx^{\prime},Tx^{\prime\prime})\Big{|}\ell_{\frac{p}{1-\theta}}(\mathbb{R}\times
Y\times Y)\right\|\leq
C\cdot\left\|(\sigma,x^{\prime},x^{\prime\prime})\Big{|}\delta_{s,\theta}^{L}(\mathbb{R}\times
X\times X)\right\|.$ (32)

for arbitrary finite sequences $x^{\prime}$, $x^{\prime\prime}$ in $X$ and
$\sigma$ in $\mathbb{R}$. Let us denote by
$\Pi^{L}_{\left(p,s,\theta\right)}(X,Y)$ the class of all Lipschitz
$\left(p,s,\theta\right)$-summing maps from $X$ to $Y$ with

$\pi^{L}_{\left(p,s,\theta\right)}(T)=\inf C.$

where the infimum is taken over all constant $C$ satisfying (32). The proof of
the following result is not difficult to prove it.

######  Proposition 25.

$\left[\Pi^{L}_{\left(p,s,\theta\right)},\pi^{L}_{\left(p,s,\theta\right)}\right]$
is a nonlinear ideal.

######  Remark 8.

  2. $\bf(1)$

If $\theta=0$, then the class $\Pi^{L}_{\left(p,s,\theta\right)}(X,Y)$
coincides with the class $\Pi^{L}_{\left(p,s\right)}(X,Y)$ which considered in
[16] for $\infty\geq p\geq q>0$ and [7] for $1\leq q<p$.

  3. $\bf(2)$

For a special case, if $p=s$, a Lipschitz $\left(p,\theta\right)$-summing map
defined in [4] if there is a constant $C\geq 0$ such that

$\left\|(\lambda,Tx^{\prime},Tx^{\prime\prime})\Big{|}\ell_{\frac{p}{1-\theta}}(\mathbb{R}\times
Y\times Y)\right\|\leq
C\cdot\left\|(\lambda,x^{\prime},x^{\prime\prime})\Big{|}\delta_{p,\theta}^{L}(\mathbb{R}\times
X\times X)\right\|.$ (33)

for arbitrary finite sequences $x^{\prime}$, $x^{\prime\prime}$ in $X$ and
$\lambda$ in $\mathbb{R^{+}}$. Let us denote by
$\Pi^{L}_{\left(p,\theta\right)}(X,Y)$ the class of all Lipschitz
$\left(p,\theta\right)$-summing maps from $X$ to $Y$ with

$\pi^{L}_{\left(p,\theta\right)}(T)=\inf C.$

where the infimum is taken over all constant $C$ satisfying (33). The next
result is a consequence of Proposition 25.

######  Corollary 2.

$\left[\Pi^{L}_{\left(p,\theta\right)},\pi^{L}_{\left(p,\theta\right)}\right]$
is a nonlinear ideal.

As a consequence of a general definition of Lipschitz interpolative ideal
procedure between metric spaces the Lipschitz $(p,\theta)$-summing map has
following characterize result.

######  Theorem 4.

[4] Let $1\leq p<\infty$, $0\leq\theta<1$ and $T\in\mathrm{Lip}(X,Y)$. The
following statements are equivalent.

    1. (i)

$T\in\Pi_{p,\theta}^{L}(X,Y)$.

    2. (ii)

There is a constant $C\geq 0$ and a regular Borel probability measure $\mu$ on
$B_{X^{\\#}}$ such that

$d_{Y}(Tx^{\prime},Tx^{\prime\prime})\leq
C\left(\int\nolimits_{B_{X^{\\#}}}\left(|fx^{\prime}-fx^{\prime\prime}|^{1-\theta}d_{X}(x^{\prime},x^{\prime\prime})^{\theta}\right)^{\frac{p}{1-\theta}}d\mu\left(f\right)\right)^{\frac{1-\theta}{p}}$

for all $x^{\prime},x^{\prime\prime}\in X$.

    3. (iii)

There is a constant $C\geq 0$ such that for all
$(x^{\prime}_{i})_{j=1}^{m},(x^{\prime\prime}_{j})_{j=1}^{m}$ in $X$ and all
$(a_{j})_{j=1}^{m}\subset\mathbb{R}^{+}$ we have

$\begin{array}[]{l}\displaystyle\left(\sum_{j=1}^{m}a_{i}d_{Y}(T(x^{\prime}_{j}),T(x^{\prime\prime}_{j}))^{\frac{p}{1-\theta}}\right)^{\frac{1-\theta}{p}}\\\
\displaystyle\leq C\underset{f\in
B_{X^{\\#}}}{\sup}\left(\sum_{j=1}^{m}a_{j}\left(|f(x^{\prime}_{j})-f(x^{\prime\prime}_{j})|^{1-\theta}d_{X}(x^{\prime}_{j},x^{\prime\prime}_{j})^{\theta}\right)^{\frac{p}{1-\theta}}\right)^{\frac{1-\theta}{p}}\text{.}\end{array}$

    4. (iv)

There exists a regular Borel probability measure $\mu$on $B_{X^{\\#}}$ and a
Lipschitz operator $v:X_{p,\theta}^{\mu}\rightarrow Y$ such that the following
diagram commutes

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{\delta_{X}}$$\textstyle{Y}$$\textstyle{\delta_{X}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\circ
i}$$\textstyle{X_{p,\theta}^{\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$

Furthermore, the infimum of the constants $C\geq 0$ in $(2)$ and $(3)$ is
$\pi_{p,\theta}^{L}\left(T\right)$.

  4. $\bf(2)$

Lipschitz $\left(s;q,\theta\right)$-mixing operators

D. Achour, E. Dahia and M. A. S. Saleh [3] defined a Lipschitz
$\left(s;q,\theta\right)$-mixing operator if there is a constant $C\geq 0$
such that

$\mathfrak{m}_{(s;q)}^{L,\theta}(\sigma,Tx^{\prime},Tx^{\prime\prime})\leq
C\cdot\delta_{q\theta}^{L}(\sigma,x^{\prime},x^{\prime\prime})$ (34)

for arbitrary finite sequences $x^{\prime}$, $x^{\prime\prime}$ in $X$ and
$\sigma$ in $\mathbb{R}$. Let us denote by
$\mathbf{M}_{(s;q)}^{L,\theta}(X,Y)$ the class of all Lipschitz
$\left(s;q,\theta\right)$-mixing maps from $X$ to $Y.$ In such case, we put

$\mathbf{m}_{(s;q)}^{L,\theta}(T)=\inf C,$

where the infimum is taken over all constant $C$ satisfying (34). The proof of
the following result is not difficult to prove it.

######  Proposition 26.

$\left[\mathbf{M}_{(s;q)}^{L,\theta},\mathbf{m}_{(s;q)}^{L,\theta}\right]$ is
a nonlinear ideal.

######  Remark 9.

Definition 6 can be generalized as follows. Let $0\leq\theta<1$. A Lipschitz
map $T$ from $X$ into $Y$ belongs to
$\left(\mathscr{A}^{L},\mathscr{B}^{L}\right)_{\theta}(X,Y)$ if there exist a
constant $C\geq 0$, a metric spaces $Z_{1}$, $Z_{2}$ and a Lipschitz maps
$S_{1}\in\mathscr{A}^{L}(X,Z_{1})$ and $S_{2}\in\mathscr{B}^{L}(X,Z_{2})$ such
that

$d_{Y}(Tx^{\prime},Tx^{\prime\prime})\leq C\cdot
d_{Z_{1}}(S_{1}x^{\prime},S_{1}x^{\prime\prime})^{1-\theta}\cdot
d_{Z_{2}}(S_{2}x^{\prime},S_{2}x^{\prime\prime})^{\theta},\ \
\forall\>x^{\prime},\;x^{\prime\prime}\in X.$ (35)

For each $T\in\left(\mathscr{A}^{L},\mathscr{B}^{L}\right)_{\theta}(X,Y)$, we
set

$\left(\mathbf{A}^{L},\mathbf{B}^{L}\right)_{\theta}(T):=\inf
C\cdot\mathbf{A}^{L}(S_{1})^{1-\theta}\cdot\mathbf{B}^{L}(S_{2})^{\theta}$
(36)

where the infimum is taken over all Lipschitz operators $S_{1}$, $S_{2}$
admitted in (35). The nonlinear ideal
$\left[\mathscr{A}^{L}_{\theta},\mathbf{A}^{L}_{\theta}\right]$ now appear as
$\left[\left(\mathscr{A}^{L},\mathscr{L}\right)_{\theta},\left(\mathbf{A}^{L},\operatorname{Lip}(\cdot)\right)_{\theta}\right]$.

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