# Spectral theory of infinite dimensional dissipative Hamiltonian systems

C. Mehl 333 Institut für Mathematik, MA 4-5, Technische Universität Berlin,
Straße des 17\. Juni 136, D-10623 Berlin, Germany.
<EMAIL_ADDRESS> 11footnotemark: 1 V. Mehrmann333
Institut für Mathematik, MA 4-5, Technische Universität Berlin, Straße des
17\. Juni 136, D-10623 Berlin, Germany<EMAIL_ADDRESS>berlin.de.  11footnotemark: 1 M. Wojtylak 222Instytut Matematyki, Wydział
Matematyki i Informatyki, Uniwersytet Jagielloński, Kraków, ul. Łojasiewicza
6, 30-348 Kraków, Poland<EMAIL_ADDRESS>(corresponding author).
44footnotemark: 4

###### Abstract

The spectral theory for operator pencils and operator differential-algebraic
equations is studied. Special focus is laid on singular operator pencils and
three different concepts of singularity of operator pencils are introduced.
The concepts are analyzed in detail and examples are presented that illustrate
the subtle differences. It is investigated how these concepts are related to
uniqueness of the underlying algebraic-differential operator equation, showing
that, in general, classical results known from the finite dimensional case of
matrix pencils and differential-algebraic equations do not prevail. The
results are then studied in the setting of structured operator pencils arising
in dissipative differential-algebraic equations. Here, unlike in the general
infinite-dimensional case, the uniqueness of solutions is closely related to
the singularity of the pencil.

Keywords. operator differential-algebraic equation, singular operator pencil,
regular operator pencil, dissipative Hamiltonian equation

AMS subject classification 2020. 37l05, 37l20, 47d06, 93b28, 93c05

##  1 Introduction

###  1.1 The setting

For regular and singular linear finite dimensional systems of differential-
algebraic equations (DAEs) $E\frac{d}{dt}x(t)=Ax(t)$, with real or complex
$n\times m$ matrices $E,A$, the spectral theory is well established through
the Kronecker canonical form of the matrix pencil $\lambda E-A$, see [32]. In
particular, the singular part of the Kroncecker canonical form ([23]) gives an
exact description of the existence and uniqueness of solutions of the Cauchy
problem $E\frac{d}{dt}x(t)=Ax(t)$, $x(0)=x_{0}$.

In this paper we study the infinite-dimensional case and consider _linear
operator differential-algebraic equations and the Cauchy problem,_

$E\frac{d}{dt}x(t)=Ax(t),\quad x(0)=x_{0}\ \ $ (1)

where the operator $E$ is a bounded operator from a Hilbert spaces
$\mathcal{X}$ to a Hilbert space $\mathcal{Y}$,
$A:\mathcal{D}(A)\subset\mathcal{X}\to\mathcal{Y}$ is densely defined and
closed, and $x_{0}\in\mathcal{X}$. The variable $t$ will always stand for a
real parameter. While in recent years several papers on this topic have
appeared, see, e.g., [15, 16, 20, 21, 26, 36, 48, 50, 55], a systematic theory
of singular operator pencils (i.e. infinite dimensional versions of Kronecker
singular blocks) is still not well developed. In the present publication we
fill this gap. In particular, we will be interested in the influence of the
spectral properties of the operator pencil $\lambda E-A$ on the uniqueness of
the solutions of (1). The systematic investigation of singular operator
pencils leads to better understanding of general linear operator differential-
algebraic equations, similarly as the theory of singular linear pencils is an
important component of the theory of differential-algebraic equations and
their numerical solution, see [32]. We show that this strategy is particularly
useful in the case when additional structure is available in the operators
$A,E$. The class of operator pencils that we consider is that arising from the
important class of _dissipative Hamiltonian differential-algebraic equations_
, i.e., equations of the form

$E\frac{d}{dt}x(t)=BQx(t),\ \ x(0)=x_{0},$ (2)

where $B$ is a dissipative operator.

This class of operator DAEs is arising in energy based modeling via port-
Hamiltonian systems in almost all physical domains such as elasticity,
electromagnetism, fluid dynamics, structural mechanics, geomechanics,
poroelasticity, gas or water transport, see e.g. [2, 3, 4, 5, 6, 17, 19, 27,
29, 33, 35, 37, 38, 39, 40, 41, 46, 47, 49, 56, 57]. An important feature of
(2) is the existence of an energy function (_Hamiltonian_) $\mathcal{H}(x)$.
This is often a quadratic function and in the case of (3) given by

$\mathcal{H}(x)=\frac{1}{2}\langle Ex,Qx\rangle,$

where $\langle\cdot,\cdot\rangle$ is an inner product on $\mathcal{Y}$. A
special case of this equation is of the form

$E\frac{d}{dt}x(t)=(J-R)x(t),\ x(t_{0})=x_{0},$ (3)

where $\mathcal{Y}=\mathcal{X}$, $B=J-R$, with $J=-J^{*}$ skew-adjoint, and
$E=E^{*}$, $R=R^{*}$ self-adjoint and positive semidefinite, and $Q$ being the
identity. The corresponding structured analysis has been a research topic of
great interest, see [1, 24, 42, 43, 44], with surprising spectral properties
compared to the general setting. The analysis of these operator differential-
algebraic equations is an active research area, see e.g. the recent papers
[21, 25, 26, 45]. However, the main assumption in all these papers is the
nonemptyness of the _resolvent set_ , i.e., the existence of
$\lambda_{0}\in\mathbb{C}$ for which $\lambda_{0}E-A$ is boundedly invertible.
The simple example $A=E=\mbox{\rm diag}(1,1/2,1/3,\dots)$ in the space
$\ell^{2}$ shows that this assumption is not necessary for existence and
uniqueness of solutions of (3), cf. Example 41.

In the following subsection we illustrate the need for a theory of singular
operator pencils with a few examples.

###  1.2 Examples

We begin with a basic example which plays a crucial role in the first part of
the paper.

###### Example 1

In the finite dimensional setting the canonical pencil having a right singular
chain is a singular Kronecker block

$\lambda E-A=\left[\begin{array}[]{cccc}\lambda&-1&&\\\
&\mathinner{\mathchoice{{\mkern 1.0mu\raise 7.00043pt\vbox{\kern
7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\scriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptscriptstyle\ldotp$}}\mkern
2.0mu\raise 4.00024pt\hbox{$\scriptscriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptscriptstyle\ldotp$}\mkern
1.0mu}}}&\mathinner{\mathchoice{{\mkern 1.0mu\raise 7.00043pt\vbox{\kern
7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\scriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptscriptstyle\ldotp$}}\mkern
2.0mu\raise 4.00024pt\hbox{$\scriptscriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptscriptstyle\ldotp$}\mkern 1.0mu}}}&\\\
&&\lambda&-1\end{array}\right].$

It is known that the corresponding differential-algebraic equation
$E\dot{x}=Ax$, $x(t_{0})=x_{0}$ does not have a unique solution.

In the first part of the paper, when dealing with general unstructured
operator pencils, generalizations of the above pencil will be studied that
satisfy

$Ee_{j}=\alpha_{j}e_{j},\quad Ae_{j}=\beta_{j}e_{j-1}.$

Here $e_{j}$ are canonical basis vectors of the space $\ell^{2}$ or
$\ell^{2}(\mathbb{Z})$ and $\alpha_{j}$, $\beta_{j}$ are complex scalars. Note
that both operators $E$ and $A$ in this case may be unbounded, however they
are both closed and densely defined. We will use this construction as a main
source of counterexamples connected with operator differential-algebraic
equations, see Examples 17, 18 and 40.

While Example 1 is a toy-example, we next present two realistic examples for
the (2) setting. Both of them, in an abstract form, will be studied in greater
detail in Section 5.3.

###### Example 2

Equations of the form (3) arise, e.g., in fluid dynamics of incompressible
Newtonian fluids, see [7, 51, 53, 54]. Let $v$ and $p$ denote the velocity and
pressure, respectively, considered as abstract functions, mapping the time
$t\in\mathbb{R}$ into appropriate spatial function spaces, [20]. The leading
order terms in the linearization of the instationary incompressible _Navier-
Stokes equations_

$\displaystyle\partial_{t}{v}-\nu\Delta v+(v\cdot\nabla)v+\nabla p$
$\displaystyle=f\qquad\text{in }\Omega\times\mathbb{R},$
$\displaystyle\nabla^{T}v$ $\displaystyle=0\qquad\text{in
}\Omega\times\mathbb{R},$

with constant density $\rho=1$ and viscosity $\nu=1$ in the limit of the
Reynolds number going to $0$ leads to the instationary _Stokes equation_

$\displaystyle\partial_{t}{v}-\Delta v+\nabla p$
$\displaystyle=f\qquad\text{in }\Omega\times\mathbb{R},$
$\displaystyle\nabla^{T}v$ $\displaystyle=0\qquad\text{in
}\Omega\times\mathbb{R}.$ (4)

Omitting the functional analytic details, see e.g. [20], then formally

$E:(v,p)\mapsto(v,0),\ J:(v,p)\mapsto(-\nabla p,-\nabla^{T}v),\
R:(v,p)\mapsto(\Delta v,0),$ (5)

and it is clear that $E$ has as kernel the functions $(0,p)$ and that the
pressure is only determined up to a constant. The associated Hamiltonian in
this example is $\frac{1}{2}\langle v,v\rangle$ and does not depend on the
pressure. The spectral theory of the associated operator and its use in the
analysis and construction of numerical methods is currently an important
research topic, see e.g. [34, 51] and the references therein.

###### Example 3

The analysis of linear poroelasticity in a bounded Lipschitz domain
$\Omega\subseteq\mathbb{R}^{d}$ with $d\in\\{2,3\\}$ (and boundary
$\partial\Omega$) was introduced in [10], see also [52]. In [2] a dissipative
Hamiltonian mixed weak formulation has been derived for the averaged
displacement field $u$ and its time derivative $w$, as well as the averaged
pressure $p$ that satisfy

$\displaystyle\begin{bmatrix}Y&0&0\\\ 0&A_{0}&0\\\
0&0&M\end{bmatrix}\begin{bmatrix}\dot{w}\\\ \dot{u}\\\
\dot{p}\end{bmatrix}=\begin{bmatrix}0&-A_{0}&D^{*}\\\ A_{0}&0&0\\\
-D&0&-K\end{bmatrix}\begin{bmatrix}w\\\ u\\\
p\end{bmatrix}+\begin{bmatrix}f\\\ 0\\\ g\end{bmatrix}.$ (6)

In the original formulation the three operators $Y$, $A_{0}$, and $M$ on the
left-hand side are self-adjoint and positive definite and the Hamiltonian
associated with this system is given by

$\mathcal{H}(w,u,p)\vcentcolon=\frac{1}{2}\,\Big{(}\langle Yw,w\rangle+\langle
A_{0}u,u\rangle+\langle Mp,p\rangle\Big{)},$ (7)

where $\tfrac{1}{2}\langle Yw,w\rangle$ describes the kinetic and
$\tfrac{1}{2}\langle Au,u\rangle+\tfrac{1}{2}\langle Mp,p\rangle$ the
potential energy. When going over to the quasi-stationary solution one sets
$Y=0$.

We refer the reader to [45] for further applications in the model class (2).

###  1.3 An overview of the results

The paper is organized as follows. In Section 2 we introduce the notation and
present some preliminary results concerning the finite and infinite spectrum
of operator pencils.

Section 3 presents the general theory for singular operator pencils. While the
situation in a finite dimensional space is clear, it appears that there is no
natural analogue in infinite dimension. In particular, the following
conditions are equivalent in finite dimensions, while essentially different in
the infinite dimensional situation:

  1. (a)

the set of points for which $\lambda E-A$ is invertible is empty;

  2. (b)

has a right or left singular polynomial: $(\lambda E-A)p(\lambda)=0$ or
$(\lambda E^{*}-A^{*})p(\lambda)=0$ for some vector valued polynomial $p$;

  3. (c)

has a right or left analytic holomorphic function $(\lambda E-A)x(\lambda)=0$
or $(\lambda E^{*}-A^{*})x(\lambda)=0$ for some vector valued holomorphic
function $x$;

  4. (d)

has a right or left approximative sequence of singular polynomials.

To deal with the situation in Section 3.1 we state five natural conditions
that a potential definition of singularity should satisfy: it should extend
the finite dimensional definition, it should imply that there are no regular
points, and it should be invariant under taking reversal pencils and
congruences, and finally, the orthogonal sum of two regular pencils should be
regular. In subsequent Sections 3.2–3.6 we show that only (b) and (d) satisfy
these requirements.

In Section 4 we study the relation between (a)–(d) and the (non)-uniqueness of
solutions of the operator DAE (1). Our main result in Theorem 36 is the
derivation of a sufficient condition for non-uniqueness of the solutions. From
this result it then follows that having a right singular chain implies non-
uniqueness of the solutions, but not conversely. The remaining concepts (a),
(c), and (d) seem to be unrelated to the question of (non)-uniqueness of
solution in general, see examples in Section 4.

While the general situation seems initially unfavourable, it becomes
dramatically different when considered in the dissipative Hamiltonian (2)
setting. In Section 5 we give a complete characterization for the uniqueness
of solutions, see Theorems 49 and (44). Several illuminative examples
throughout the paper illustrate the results.

##  2 Preliminaries

###  2.1 Basic notations

In this paper we consider the spectral theory of differential-algebraic
equations (DAEs) with coefficients that are operators in Hilbert spaces. The
scalar product and norm will be denoted respectively by
$\left<\cdot,\cdot\right>$ and $\left\|\cdot\right\|$. It will be always clear
from the context in which space the norm is taken. In several examples we will
use the spaces $\ell^{2}=\ell^{2}(\mathbb{N})$ and $\ell^{2}(\mathbb{Z})$ of
square-summable sequences. For the general theory of unbounded operators, only
briefly presented below, we refer the reader e.g. to [58].

By a linear operator (or in short: operator) we understand a linear mapping
$S:\mathcal{D}(S)\to\mathcal{Y}$, where the _domain_ $\mathcal{D}(S)$ is a
linear subspace of $\mathcal{X}$. By $\mathbf{B}(\mathcal{X},\mathcal{Y})$ we
denote the _set of bounded linear operators_ from $\mathcal{X}$ to
$\mathcal{Y}$. If $S$ is a closed, densely defined operator, then we say that
it is _invertible_ , if there exists an operator
$T\in{\mathbf{B}}(\mathcal{Y},\mathcal{X})$ such that $TS=I_{\mathcal{D}(S)}$
and $ST=I_{\mathcal{X}}$, where $I_{\mathcal{Z}}$ denotes the identity
operator on the space $\mathcal{Z}$.

For an operator $A:\mathcal{D}(A)\to\mathcal{Y}$ we define the _adjoint_
operator $A^{*}$ as usually, i.e., we set
$\mathcal{D}\left(A\right)^{*}=\\{y\in\mathcal{Y}:\text{there exists
}z\in\mathcal{X}\ \langle y,Ax\rangle=\langle z,x\rangle,\
x\in\mathcal{D}(A)\\}$ and $A^{*}y$ equals by definition the (unique) $z$ from
the previous formula. We will use without saying the fact that $A$ is closable
if and only if $A^{*}$ is densely defined. In particular, in the space
$\ell^{2}(\Gamma)$, where $\Gamma\in\\{\mathbb{N},\mathbb{Z}\\}$, every
operator for which the linear span of the standard orthogonal basis is
contained in $\mathcal{D}\left(A\right)\cap\mathcal{D}\left(A^{*}\right)$ is
closable and densely defined.

Although our focus will be on operator pencils associated with the operator
DAEs (ODAEs) (1) and (3), we will also deal with more general _operator
pencils_ of the form

$P(\lambda)=\lambda E-A,$ (8)

with $E:\mathcal{D}(E)\to\mathcal{Y}$, $A:\mathcal{D}(A)\to\mathcal{Y}$ being
closed and densely defined and such that $P(\lambda)$ is closed and densely
defined for all $\lambda\in\mathbb{C}$. This happens, in particular, if one of
the operators $E$ or $A$ is bounded, however this is not the only possibility.
We will always use the symbol $\lambda$ for the free variable, therefore by
$P(\lambda)$ will always mean the operator pencil in (8). When we evaluate
that pencil at a specific complex number $\lambda_{0}\in\mathbb{C}$ then
$P(\lambda_{0})$ denotes an operator from $\mathcal{D}(A-\lambda_{0}E)$ to
$\mathcal{Y}$.

When $\mathcal{X}_{j}$ ($j=1,2$) are Hilbert spaces, then by
$\mathcal{X}_{1}\oplus\mathcal{X}_{2}$ we denote their orthogonal sum, i.e.,
their Cartesian product with the (unique) norm satisfying
$\left\|(f,g)\right\|^{2}=\left\|f\right\|^{2}+\left\|g\right\|^{2}$. We will
also use the symbol $f\oplus g$ for a pair $f\in\mathcal{X}_{1}$,
$g\in\mathcal{X}_{2}$. If, additionally, $\mathcal{Y}_{j}$ ($j=1,2$) are
Hilbert spaces and $S_{j}$ are operators from
$\mathcal{D}\left(S_{j}\right)\subseteq\mathcal{X}_{j}$ to $\mathcal{Y}_{j}$
($j=1,2$), then by $S_{1}\oplus S_{2}$ we understand the operator from
$\mathcal{D}\left(S_{1}\right)\oplus\mathcal{D}\left(S_{2}\right)$ to
$\mathcal{Y}_{1}\oplus\mathcal{Y}_{2}$ given by $(S_{1}\oplus
S_{2})(f_{1}\oplus f_{2})=S_{1}f_{1}\oplus S_{2}f_{2}$. In a similar way we
denote the infinite orthogonal sum of Hilbert spaces
$\bigoplus_{j}\mathcal{X}_{j}$, see e.g. [58] for the general theory.

To develop the spectral theory of operator pencils we first define various
type of _spectra_.

###### Definition 4

Consider an operator pencil as in (8).

  1. 1)

By $s\big{(}P(\lambda)\big{)}$ we denote the _set of singular points of
$P(\lambda)$_, i.e., the set of all $\lambda_{0}\in{\mathbb{C}}$ for which the
operator $P(\lambda_{0})$ is not invertible.

  2. 2)

By $s_{\operatorname{p}}\big{(}P(\lambda)\big{)}$ we denote the _set of point
singularities of $P(\lambda)$_, i.e., values $\lambda_{0}\in\mathbb{C}$ for
which there exists
$x\in\mathcal{D}\big{(}P(\lambda_{0})\big{)}\setminus\\{0\\}$ with
$P(\lambda_{0})x=0$.

  3. 3)

By $s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$ we denote the
_approximative singularities_ of $P(\lambda)$, i.e., the set of all points
$\lambda_{0}\in{\mathbb{C}}$ for which there exists a sequence
$\\{x_{n}\\}_{n=1}^{\infty}$ of unit norm vectors
$x_{n}\in\mathcal{D}\big{(}P(\lambda_{0})\big{)}$ such that
$P(\lambda_{0})x_{n}\to 0$. We will use without saying that the sequence
$\\{x_{n}\\}_{n=1}^{\infty}$ above might be equivalently assumed to satisfy
$\,\liminf_{n\to\infty}\left\|x_{n}\right\|>0$ only, instead of being
normalized.

  4. 4)

The set of _regular points_ is defined as
$\rho\big{(}P(\lambda)\big{)}:={\mathbb{C}}\setminus
s\big{(}P(\lambda)\big{)}$.

These definitions of spectra of operator pencils are direct generalizations of
respective definitions of spectra for operators when
$\mathcal{Y}=\mathcal{X}$, $E=I_{\mathcal{X}}$. We highlight that all spectra
defined in Definition 4 are viewed as subsets of the complex plane (excluding
the point infinity). Infinity as a spectral point will be discussed in detail
in Subsection 2.3.

###  2.2 Finite spectrum of operator pencils

In this section we derive some properties of the spectra of general operator
pencils, extending some classical results for operators, see e.g. [31, Chapter
2].

###### Proposition 5

Let $P(\lambda)=\lambda E-A$ be as in (8), with
$E\in\mathbf{B}(\mathcal{X},\mathcal{Y})$ and $A$ being a closed and densely
defined operator from $\mathcal{D}\left(A\right)\subseteq\mathcal{X}$ to
$\mathcal{Y}$. Then the following statements hold:

  1. (i)

If $\lambda_{0}\in\rho\big{(}P(\lambda)\big{)}$ then the disc

$D(\lambda_{0}):=\left\\{z\in{\mathbb{C}}:|z-\lambda_{0}|<\frac{1}{\left\|(\lambda_{0}E-A)^{-1}\right\|\cdot{\left\|E\right\|}}\right\\}$

is contained in $\rho\big{(}P(\lambda)\big{)}$. In particular,
$\rho(P\big{(}\lambda)\big{)}$ is an open subset of ${\mathbb{C}}$.

  2. (ii)

If $\rho\big{(}P(\lambda)\big{)}$ is nonempty, then the boundary of
$s\big{(}P(\lambda)\big{)}$ is contained in
$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$.

  3. (iii)

The sets $s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$ and
$s\big{(}P(\lambda)\big{)}$ are closed subsets of ${\mathbb{C}}$.

Proof. (i) For $\lambda_{0}\in\rho\big{(}P(\lambda)\big{)}\cap{\mathbb{C}}$,
the formal power series

$\sum_{j=0}^{\infty}(-1)^{j}(z-\lambda_{0})^{j}(\lambda_{0}E-A)^{-1}\big{(}E(\lambda_{0}E-A)^{-1}\big{)}^{j}$

constitutes the inverse of the operator $P(z)$ for $z\in D(\lambda_{0})$.
Hence $D(\lambda_{0})$ is contained in the set of regular points of of
$P(\lambda)$.

(ii) Assume that $\lambda_{0}$ is on the boundary of
$s\big{(}P(\lambda)\big{)}$ and let
$\\{\lambda_{n}\\}_{n=1}^{\infty}\subseteq{\mathbb{C}}$ be a sequence
contained in the set of regular points of $P(\lambda)$ that is converging to
$\lambda_{0}$. The inclusion in (i) applied for each $n$ implies that

$\operatorname{dist}\Big{(}\lambda_{n},s\big{(}P(\lambda)\big{)}\Big{)}\geq\frac{1}{\left\|(\lambda_{n}E-A)^{-1}\right\|\cdot{\left\|E\right\|}}.$
(9)

As a consequence we obtain

$\left\|(\lambda_{n}E-A)^{-1}\right\|\geq\frac{1}{\operatorname{dist}\big{(}\lambda_{n},s\big{(}P(\lambda)\big{)}\big{)}\cdot\left\|E\right\|}\geq\frac{1}{|\lambda_{n}-\lambda_{0}|\cdot\left\|E\right\|}\to\infty,\quad
n\to\infty.$

Hence, there exists a sequence
$\\{f_{n}\\}_{n=1}^{\infty}\subseteq\mathcal{Y}$ with $\left\|f_{n}\right\|=1$
and $\left\|(\lambda_{n}E-A)^{-1}f_{n}\right\|\to\infty$. Then setting
$g_{n}:=(\lambda_{n}E-A)^{-1}f_{n}$ we have
$g_{n}\in\mathcal{D}(\lambda_{n}E-A)=\mathcal{D}(A)$ and

$\left\|(\lambda_{0}E-A)\frac{g_{n}}{\left\|g_{n}\right\|}\right\|\leq|\lambda_{0}-\lambda_{n}|\left\|E\right\|+\frac{\left\|(\lambda_{n}E-A)g_{n}\right\|}{\left\|g_{n}\right\|}=|\lambda_{0}-\lambda_{n}|\left\|E\right\|+\frac{\left\|f_{n}\right\|}{\left\|g_{n}\right\|}\to
0.$

Then, by definition, $\lambda_{0}\in
s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$.

(iii) Let $\\{\lambda_{n}\\}_{n=1}^{\infty}\subseteq
s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$ converge to some
$\lambda_{0}\in{\mathbb{C}}$. Let also $\\{x_{n}\\}_{n=1}^{\infty}$ be a
sequence of unit norm vectors in $\mathcal{D}(A)$ such that
$\left\|(\lambda_{n}E-A)x_{n}\right\|\leq 1/n$ for $n\geq 1$. Such a sequence
exists due to $\lambda_{n}\in s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$.
Since $E$ is bounded it follows that

$\left\|(\lambda_{0}E-A)x_{n}\right\|\leq\left\|(\lambda_{n}E-A)x_{n}\right\|+|\lambda_{n}-\lambda_{0}|\cdot\left\|Ex_{n}\right\|\to
0,\quad n\to\infty,$

which shows that $\lambda_{0}\in
s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$. The closedness of
$s\big{(}P(\lambda)\big{)}$ is clear by (i).

In this subsection we have reviewed several classical spectral concepts for
linear operators and extended them to operator pencils. In the next subsection
we discuss the spectrum at $\infty$.

###  2.3 Infinity as a spectral point, the reversal pencil

In this section we use the concept of the reversal of an operator pencil to
define and analyze the spectral properties at $\infty$.

###### Definition 6

_The reversal of an operator pencil $\lambda E-A$_ is defined as the pencil
$\lambda A-E$. Further, we say that $\infty$ is a _point singularity
(approximative singularity, singularity, regular point)_ of $P(\lambda)$ if
zero belongs to the set of point singularities, (respectively approximative
singularities, singularities, regular points) of the reversal $\lambda A-E$.

It is clear that $\lambda_{0}\in\mathbb{C}\cup\\{\infty\\}$ is a singularity
(approximative singularity, singularity, regular point) of $\lambda E-A$ if
and only if $\lambda_{0}^{-1}$ is a singularity (approximative singularity,
singularity, regular point, respectively) of $\lambda A-E$, regardless whether
the operators $A$ or $E$ are bounded or not, with the standard conventions
$\infty^{-1}=0$ and $0^{-1}=\infty$. With this definition it is also tempting
to consider the different singularities of the pencil as subsets of the
extended complex plane $\mathbb{C}\cup\\{\infty\\}$. Unfortunately, however,
Proposition 5 is not true if $\mathbb{C}$ is replaced by
$\mathbb{C}\cup\\{\infty\\}$, as the following Example 7 shows. This example
also shows simultaneously that Proposition 5 is not true if $A$ is assumed to
be bounded instead of $E$.

###### Example 7

Let $A$ be a closed densely defined operator with spectrum equal to the whole
complex plane, e.g., let $A$ be the multiplication operator by $z$ in
$L^{2}(\mu)$, where $\mu$ is some finite measure supported on the whole
complex plane. Thus the operator pencil $\lambda I-A$ has also the spectrum
equal to the whole complex plane. The reversal $Q(\lambda)=\lambda A-I$,
however, has zero in the set of its regular points as $Q(0)=I$. Therefore,
infinity is a regular point for $P(\lambda)$ and the set of singular points is
not closed in the extended complex plane $\mathbb{C}\cup\\{\infty\\}$.
Similarly, the set of singular points of $Q(\lambda)$ equals
$\mathbb{C}\setminus\\{0\\}$, which is not closed in $\mathbb{C}$.

Therefore, in the following we keep viewing $s\big{(}P(\lambda)\big{)}$,
$s_{\operatorname{p}}\big{(}P(\lambda)\big{)}$,
$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$,
$\rho\big{(}P(\lambda)\big{)}$ as subsets of the complex plane, regardless on
the behaviour of the pencil at infinity. A detailed study of infinity as a
spectral point in the case that the resolvent set is nonempty is presented in
the recent paper [21].

##  3 Singular operator pencils - general theory

The main aim of our paper is to study the situation when the set of regular
points of an operator pencil (8) is empty, which can happen for several
reasons. Distinguishing between these different reasons requires a definition
of a _singular operator pencil_ , see e.g. [45] for some suggestions for
possible definitions. In order to see which definition is most appropriate for
a certain purpose, let us create a list of criteria for a good definition.

###  3.1 Criteria for a definition of a singular operator pencil

We suggest that a definition of singularity for an operator pencil should meet
the following criteria.

  1. 1)

In a finite dimensional situation it reduces to the standard definition.

  2. 2)

It implies that there are no regular points.

  3. 3)

A pencil is singular if and only if its reversal is singular. Hence infinity
is not a regular point. In particular, no pencil of the form $\lambda I-A$
should be called singular.

  4. 4)

If the pencils $\lambda E_{j}-A_{j}$ ($j=1,2)$ are not singular, then neither
is $(\lambda E_{1}-A_{2})\oplus(\lambda E_{2}-A_{2})$.

  5. 5)

A transformation $S(\lambda E-A)T$, with $S,T$ invertible, keeps the pencil
singular.

While the criteria 1)–5) seem quite natural, we will see in the following that
it is not easy to have them satisfied simultaneously.

###  3.2 Emptyness of the set of regular points, initial remarks

Recall that for a general operator pencil $P(\lambda)$ as in (8) the set of
point singularities $s_{\operatorname{p}}\big{(}P(\lambda)\big{)}$ is
contained in the set of approximate singularities
$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}$, which in turn is contained in
the set of all singularities $s\big{(}P(\lambda)\big{)}$, and we discuss all
these sets as subsets of the complex plane, treating infinity separately. For
this we consider the following three important cases:

$\displaystyle s_{\operatorname{p}}\big{(}P(\lambda)\big{)}=\mathbb{C}$ and
$\displaystyle\infty\text{ is a point singularity};$ (10) $\displaystyle
s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}=\mathbb{C}$ and
$\displaystyle\infty\text{ is an approximate singularity};$ (11)
$\displaystyle s\big{(}P(\lambda)\big{)}=\mathbb{C}$ and
$\displaystyle\infty\text{ is a singularity}.$ (12)

Clearly, (10) implies (11) and this implies (12), and in the finite
dimensional case it follows from the Kronecker canonical form [23] that all
these conditions are equivalent. We now present some examples that show that
in the infinite dimensional case the converse implications do not hold in
general.

###### Example 8

Let $E$ be a bounded operator with $0$ in the approximative spectrum, but not
in the point spectrum. Then the pencil $P(\lambda)=\lambda E-E$ satisfies (11)
but not (10), as, e.g., $P(2)=E$ has a trivial kernel.

###### Example 9

Let $A$ be a closed, densely defined operator on some Hilbert space
$\mathcal{X}$ having the complex plane as its spectrum, but not as its
approximative spectrum, e.g., $A$ is a symmetric but not self-adjoint
operator. Then

$P(\lambda)=\lambda(I_{\mathcal{X}}\oplus 0)+A\oplus 1,\quad\text{in
}{\mathcal{X}}\oplus{\mathbb{C}}$ (13)

is an operator pencil which satisfies (12) but not (11).

These two examples are rather simple and do not present the nature of the
problem yet. More elaborate examples will be given subsequently.

One may wonder if any of the conditions (10)–(12) would be suitable as a
definition of singularity for operator pencils, i.e., if this definition
satisfies the requirements 1)–5) of Subsection 3.1. The following example
shows an immediate problems with item 4), see also [30] for a similar
construction.

###### Example 10

It is well-known that the adjoint $S_{0}^{\ast}$ of the unilateral shift
$S_{0}$ in $\mathcal{X}_{0}=\ell^{2}$ is a bounded linear operator with the
point spectrum being the open unit disc. Hence it is possible to find a
sequence of complex numbers $\\{\alpha_{n}\\}_{n=1}^{\infty}$ such that

$\bigcup_{n=1}^{\infty}\sigma_{p}(S_{0}^{\ast})+\alpha_{n}={\mathbb{C}}.$

Defining $\tilde{\mathcal{X}}=\bigoplus_{n=1}^{\infty}\mathcal{X}_{0}$ and
$\tilde{S}=\bigoplus_{n=1}^{\infty}(S_{0}^{\ast}+\alpha_{n}I_{\mathcal{X}_{0}})$,
we get that the point spectrum of $\tilde{S}$ equals ${\mathbb{C}}$. Then the
pencil

$P(\lambda)=\lambda E-A\quad\text{with }E=I_{\tilde{\mathcal{X}}}\oplus
0,\;A:=\tilde{S}\oplus 1$

in $\mathcal{X}=\tilde{\mathcal{X}}\oplus{\mathbb{C}}$ has the set of point
singularities equal to $\mathbb{C}$ and infinity is a point singularity as
well. Furthermore, it follows from the construction that one may represent a
singular $P(\lambda)$ as an orthogonal sum of two operator pencils, each of
them having a nonempty set of regular points. As an example to see this, let

$J_{1}=\\{n:\mbox{\rm Re}\,\alpha_{n}\leq 0\\},\quad J_{2}=\\{n:\mbox{\rm
Re}\,\alpha_{n}>0\\}.$

Setting ${\mathcal{X}_{i}}=\bigoplus_{j\in J_{i}}\mathcal{X}_{0}$ and
$S_{i}=\bigoplus_{j\in J_{i}}(S_{0}^{*}+\alpha_{j}I_{\mathcal{X}_{0}})$,
$i=1,2$, we obtain

$P(\lambda)=(\lambda I_{\mathcal{X}_{1}}-S_{1})\oplus(\lambda
I_{\mathcal{X}_{2}}-S_{2})\oplus(\lambda 0-1),$

which (by combining one of the first two summands with the last) can obviously
be reduced to two summands as required.

To obtain a definition of singular operator pencils satisfying our desired
criteria, in the next subsection we generalize the concept of singular chains
from the case of matrix pencils to operator pencils.

###  3.3 Right and left singular polynomials

In this subsection, we first review the theory of singular chains of matrix
pencils. Even in the finite-dimensional case, one has to distinguish between
right and left singular chains and we will focus on the former ones first.

We say that the $k+1$ vectors $x_{0},x_{1},\dots,x_{k}$ ($k\geq 0$) form a
right singular chain for a square complex matrix pencil $\lambda E-A$ if they
are linearly independent and satisfy

$\displaystyle Ax_{0}=0,\quad Ax_{j+1}=Ex_{j}\neq 0,\ j=0,\dots k-1,\quad
Ex_{k}=0.$ (14)

Note that for $k=0$ condition (14) just means that $\ker A\cap\ker
E\neq\left\\{0\right\\}$. Moreover, having a right singular chain directly
implies (10). Indeed, for any $\lambda_{0}\in{\mathbb{C}}$ we can define

$x(\lambda_{0}):=\sum_{j=0}^{k}\lambda_{0}^{j}x_{j}$

and, due to the linear independence of $x_{0},\dots,x_{k}$, we have that
$x(\lambda_{0})\neq 0$ and

$\displaystyle(\lambda_{0}E-A)x(\lambda_{0})$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}\lambda_{0}^{j+1}Ex_{j}-\sum_{j=0}^{k}\lambda_{0}^{j}Ax_{j}$
(15) $\displaystyle=$
$\displaystyle\lambda_{0}^{k+1}Ex_{k}+\sum_{j=0}^{k-1}\lambda_{0}^{j+1}(Ex_{j}-Ax_{j+1})+Ax_{0}=0.$
(16)

Furthermore, $\infty$ is a spectral point of $\lambda E-A$, because $E$ has a
nontrivial kernel.

Also let us recall, that if $x_{0},\dots,x_{k}$ satisfy (14) but are not
linearly independent or do not satisfy $Ex_{j}\neq 0$ for all $j=0,\dots,k-1$,
then a shorter right singular chain is contained in the span of
$x_{0},\dots,x_{k}$.

In a similar way one defines left singular chains as right singular chains of
the conjugate transpose pencil $\lambda E^{*}-A^{*}$. Note that every square
matrix pencil that has a left singular chain also necessarily possesses a
right singular chain, although their lengths may be different.

Finally, a matrix pencil is called _singular_ if it possesses either a left or
a right singular chain. From the discussion above, it is apparent that one may
equivalently define singular pencils as those for which there exists a nonzero
vector valued polynomial $p(\lambda)$ with either
$(\lambda_{0}E-A)p(\lambda_{0})=0$ for all $\lambda_{0}\in\mathbb{C}$ or
$p^{*}(\lambda_{0})(\lambda_{0}E-A)=0$ for all $\lambda_{0}\in\mathbb{C}$. In
view of these observations we introduce the following definition for operator
pencils.

###### Definition 11

Consider an operator pencil of the form (8). We say that an
$\mathcal{X}$-valued polynomial $p(\lambda)$ is _a right singular polynomial
for the pencil $\lambda E-A$_ if $p(\lambda)$ is a nonzero polynomial,
$p(\lambda_{0})\in\mathcal{D}\left(E\right)\cap\mathcal{D}\left(A\right)$ and
$(\lambda_{0}E-A)p(\lambda_{0})=0$ for every $\lambda_{0}\in\mathbb{C}$. We
say that an $\mathcal{Y}$-valued polynomial $q(\lambda)$ is a left singular
polynomial for the pencil $\lambda E-A$ if it is a right singular polynomial
for the adjoint pencil $\lambda E^{*}-A^{*}$. We call a pencil _point
singular_ if has either a left or a right singular polynomial.

In the following we show that Definition 11 satisfies the singularity criteria
1)–5) from Subsection 3.1. For this we will need the following lemma.

###### Lemma 12

Consider an operator pencil of the form (8). If the pencil $\lambda E-A$ has a
right singular polynomial, then there exists a right singular polynomial
$p_{0}(\lambda)$ such that both $p_{0}(\lambda)$ and
$\operatorname{rev}p_{0}(\lambda)$ do not have any roots in the complex plane.
Furthermore, $\operatorname{rev}p_{0}(\lambda)$ is a right singular polynomial
for $\lambda A-E$.

Proof. Suppose that $p(\lambda)=\sum_{j=0}^{k}\lambda^{j}a_{j}$
($a_{1},\dots,a_{k}\in\mathcal{X}$) is a right singular polynomial of $\lambda
E-A$. As the linear span of $a_{1},\dots,a_{k}$ is finite-dimensional, we can
rewrite $p(\lambda)$ as $p(\lambda)=\sum_{j=0}^{l}p_{j}(\lambda)f_{j}$, with
some scalar polynomials $p_{1}(\lambda),\dots,p_{l}(\lambda)$, orthonormal
vectors $f_{1},\dots,f_{l}$ and $l\leq k$. Let $q(\lambda)$ be the greatest
common divisor of $p_{1}(\lambda),\dots,p_{l}(\lambda)$. We set

$p_{0}(\lambda)=\frac{1}{q(\lambda)}{p(\lambda)}=\sum_{j=1}^{l}\frac{p_{j}(\lambda)}{q(\lambda)}f_{j}.$

Observe that $p_{0}(\lambda)$ is an $\mathcal{X}$-valued polynomial, which has
no zeros in the complex plane and is a right singular polynomial of $\lambda
E-A$ due to $(A-\lambda
E)\frac{p(\lambda)}{q(\lambda)}=\frac{1}{q(\lambda)}(A-\lambda
E){p(\lambda)}=0$. By properties of the reversal polynomial
$\operatorname{rev}p_{0}(\lambda)$ has no zeros in the complex plane as well
and is a right singular polynomial for the reversal pencil $\lambda A-E$.

We now show that a point singular operator pencil has no regular points.

###### Proposition 13

Consider an operator pencil $P(\lambda)=\lambda E-A$ with
$E\in\mathbf{B}(\mathcal{X},\mathcal{Y})$ and $A$ being a closed and densely
defined operator from $\mathcal{D}\left(A\right)\subseteq\mathcal{X}$ to
$\mathcal{Y}$. Then the following statements hold:

  1. (i)

If $\lambda E-A$ has a right singular polynomial $p(\lambda)$, then the set of
point singularities coincides with the complex plane and also $\infty$ is a
point singularity, i.e., (10) holds.

  2. (ii)

If $\lambda E-A$ has a left singular polynomial, then the set of all
singularities of $\lambda E-A$ is the whole complex plane and also $\infty$ is
a singularity, i.e., (12) holds.

Proof. (i) The first statement is obvious by Lemma 12. Part (ii) follows
directly from the fact that if an operator $Z^{*}$ is not invertible, then
neither is $Z$.

We now have the following observations concerning the singularity criteria
from Subsection 3.1. The concept of point singularity in Definition 11
obviously satisfy criteria 1) and 5). Criterion 2) follows from Proposition
13, and criterion 3) from Lemma 12. Criterion 4) follow by contraposition from
the fact that if

$\big{(}(\lambda E_{1}-A_{1})\oplus(\lambda E_{2}-A_{2})\big{)}\
\big{(}p_{1}(\lambda)\oplus p_{2}(\lambda)\big{)}=0,$

then $p_{j}(\lambda)$ is either the zero polynomial or a right singular
polynomial for $\lambda E_{j}-A_{j}$ $(j=1,2)$.

###### Example 14

The operator pencil $\lambda E-A$ of Example 10 has $\mathbb{C}$ as set of
point singularities and infinity is a point singularity as well. However, the
pencil is not point singular according to Definition 11. Indeed, from the
construction it follows that it can be decomposed into two pencils with
nonempty sets of regular points.

Although having a right or left singular polynomial is in accordance with our
list of singularity criteria, in the infinite dimensional context it is rather
restrictive. In the following subsections we will therefore discuss other
concepts.

###  3.4 Right and left singular holomorphic functions

An obvious generalization of left and right singular polynomials is obtained
via the transition to holomorphic functions. However, we will see in this
subsection that they are not suitable as a concept describing singularity of
an operator pencil.

###### Definition 15

Consider an operator pencil of the form (8). We say that an
$\mathcal{X}$-valued function $x(\lambda)$ is a _left singular function_ for
$\lambda E-A$ if it is defined and holomorphic on $\mathbb{C}$ except,
possibly, for a discrete set of points and

$(\lambda E-A)x(\lambda)=0,\quad x(\lambda)\neq 0,$

for all $\lambda\in\mathbb{C}$, except, possibly, for a discrete set of
points.

For completeness we state an analogue of Proposition 13, noting that for the
proof of (i) one needs to use the fact that the set of zeros of a holomorphic
function is nowhere dense in $\mathbb{C}$.

###### Proposition 16

Let $P(\lambda)=\lambda E-A$, with $E\in\mathbf{B}(\mathcal{X},\mathcal{Y})$
and $A$ being a closed and densely defined operator from
$\mathcal{D}\left(A\right)\subseteq\mathcal{X}$ to $\mathcal{Y}$. Then the
following statements hold:

  1. (i)

If $\lambda E-A$ has a right singular function $x(\lambda)$, then the set of
point singularities contains all nonzero points of the domain of the function
$x(\lambda)$. In particular, the set of approximative singularities is the
whole complex plane.

  2. (ii)

If $\lambda E-A$ has a left singular function, then the set of all
singularities of $\lambda E-A$ is the whole complex plane.

We now present two examples of pencils having right singular functions.

###### Example 17

Consider the following operators in the Hilbert space $\ell_{2}$ of square-
summable sequences:

$\lambda E-A=\left[\begin{array}[]{ccccc}\lambda&-1&&&\\\ &\lambda/2&-1&&\\\
&&\lambda/3&&\\\ &&&\mathinner{\mathchoice{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\scriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptscriptstyle\ldotp$}}\mkern
2.0mu\raise 4.00024pt\hbox{$\scriptscriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptscriptstyle\ldotp$}\mkern
1.0mu}}}&\mathinner{\mathchoice{{\mkern 1.0mu\raise 7.00043pt\vbox{\kern
7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\textstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\textstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\textstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptstyle\ldotp$}}\mkern 2.0mu\raise
4.00024pt\hbox{$\scriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptstyle\ldotp$}\mkern 1.0mu}}{{\mkern 1.0mu\raise
7.00043pt\vbox{\kern 7.00043pt\hbox{$\scriptscriptstyle\ldotp$}}\mkern
2.0mu\raise 4.00024pt\hbox{$\scriptscriptstyle\ldotp$}\mkern 2.0mu\raise
1.00006pt\hbox{$\scriptscriptstyle\ldotp$}\mkern 1.0mu}}}\end{array}\right],$

i.e., $A$ is the backward shift, and $E$ is a diagonal operator. With
$x_{j}=\frac{1}{(j-1)!}e_{j}$, $j=1,2,\dots$, where $e_{1},e_{2},\dots,$
stands for the canonical basis of $\ell^{2}$, one clearly has that

$x(\lambda)=\sum_{j=1}^{\infty}\lambda^{j}x_{j}$

is everywhere square-summable and nonzero. Furthermore,

$Ax_{1}=0\quad\mbox{and}\quad Ax_{j}=\frac{1}{(j-1)!}e_{j-1}=Ex_{j-1},\quad
j=2,3,\dots$

Therefore, $\sum_{j=1}^{\infty}\lambda^{j}Ax_{j}$ is square-summable and
$x(\lambda)$ belongs to $\mathcal{D}\left(A\right)$ for all $\lambda$.
Moreover, we have

$(\lambda
E-A)x(\lambda)=\lambda\sum_{j=1}^{\infty}\lambda^{j}Ex_{j}-\sum_{j=1}^{\infty}\lambda^{j}Ax_{j}=0,$

i.e. $x(\lambda)$ is a right singular function of $\lambda E-A$.

###### Example 18

Consider a pencil $\lambda I-A$ in the space $\ell^{2}(\mathbb{Z})$, i.e., the
Hilbert space of square-summable sequences with index set $\mathbb{Z}$, where
the operator $A$ is defined with the use of the canonical basis as follows:

$A_{0}(e_{j})=\frac{|j|!}{|j-1|!}e_{j-1},\quad j\in\mathbb{Z}.$

Clearly, $A_{0}$ is a closable, densely defined operator, and let $A$ be its
closure. Define $x(\lambda)$ by the Laurent series

$x(\lambda)=\sum_{j=-\infty}^{\infty}\frac{\lambda^{j}}{|j|!}e_{j},$

which is convergent everywhere except at zero and infinity. Furthermore,

$A\frac{e_{j}}{|j|!}=\frac{1}{|j-1|!}e_{j-1},\quad j\in\mathbb{Z},$

and therefore $\sum_{j=-\infty}^{\infty}\lambda^{j}A\frac{e_{j}}{|j|!}$ is
square-summable and $x(\lambda)$ belongs to $\mathcal{D}\left(A\right)$ for
all $\lambda\neq 0$. Moreover, $(\lambda I-A)x(\lambda)=0$ as it can be
immediately seen by shifting the summation index in $Ax(\lambda)$.

In view of Example 18, ‘having a right or left singular function’ does not
seem to be a good generalisation of the notion of a singular matrix pencil. It
satisfies the criteria 1), 2), 4) and 5) from Subsection 3.1, but it does not
satisfy criterion 3). One could think of some variations, e.g., ‘both $\lambda
E-A$ and $\lambda A-E$ have a (left or right) singular functions’ as another
definition of singularity. This would satisfy the criteria 1), 2), 3) and 5),
but no longer criterion 4) as the following example demonstrates.

###### Example 19

Let $A$ be the operator from Example 18 and consider the operator pencil

$\lambda(I\oplus A)-(A\oplus I)=(\lambda I-A)\oplus(\lambda A-I).$

Then $x(\lambda)\oplus 0$ is a right singular function for the pencil, while
$0\oplus x(\lambda)$ is a right singular function for the reversal. It should
be noted, that none of the coefficients of the pencil is bounded, however, the
pencil is closed and densely defined for all $\lambda\in\mathbb{C}$.

###  3.5 Operator pencils with approximative joint kernel

One of the situations that we would like to cover by the notion of singularity
is the case of operator pencils of the form (8) where $A$ and $E$ have an
_approximative joint kernel_ , i.e., there exists a sequence of unit norm
vectors
$\\{x_{n}\\}_{n=1}^{\infty}\subseteq\mathcal{D}\left(A\right)\cap\mathcal{D}\left(E\right)$
with $Ax_{n}\to 0$ and $Ex_{n}\to 0$ for $n\to\infty$. Clearly, if this
happens then $s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}=\mathbb{C}$ and
infinity is an approximate singularity as well. Therefore, it is tempting to
call such pencils singular. Note that this situation is not covered by the
concept of point singularity in Definition 11, unless the dimension is finite,
as the following example demonstrates.

###### Example 20

Let $\mathcal{X}=\mathcal{Y}=\ell^{2}$ and let $A=E=\mbox{\rm
diag}(1,1/2,1/3,\dots)$. Here both operators $E$ and $A$ are bounded (even
compact) and the pencil $\lambda E-E$ has a unit norm sequence $x_{n}:=e_{n}$
(the $n$-th vector of the canonical basis) with $Ax_{n}=Ex_{n}\to 0$.

Another argument for considering the case of approximative joint kernels as a
concept for a singular operator pencil is the following. Consider two
sequences of (not necessarily orthogonal) projections
$P_{n}:\mathcal{X}\to\mathcal{X}_{n}$, $Q_{n}:\mathcal{Y}\to\mathcal{Y}_{n}$
onto finite dimensional subspaces
$\mathcal{X}_{n}\subseteq\mathcal{D}\left(A\right)\cap\mathcal{D}\left(E\right)$,
$\mathcal{Y}_{n}\subseteq\mathcal{Y}$ that satisfy the following conditions:

$\displaystyle P_{n}f\to f\ \mbox{for }n\to\infty,\quad f\in\mathcal{X},$ (17)
$\displaystyle Q_{n}AP_{n}f\to Af,\ Q_{n}EP_{n}f\to Ef\ \mbox{for
}(n\to\infty),\ f\in\mathcal{D}\left(A\right)\cap\mathcal{D}\left(E\right).$
(18)

This is the typical setting in numerical methods where the solution of
partial-differential equation is approximated using some discretization
method, e.g. via Petrov-Galerkin projection in the finite element method, or
truncated Taylor or Fourier series, see e.g. [11, 12, 13, 18] and the
references therein.

In what follows we will view the operators $Q_{n}EP_{n}$ and $Q_{n}AP_{n}$ as
acting from the range of $P_{n}$ to the range of $Q_{n}$, both being finitely
dimensional (although formally these operators act from $\mathcal{X}$ to
$\mathcal{Y}$). Note that for operators from the range of $P_{n}$ to the range
of $Q_{n}$ we can define the Frobenius norm as
$\left\|X\right\|_{F}:=(\operatorname{tr}(X^{*}X))^{1/2}$. Thus, we can
measure the _distance to singularity_ of $Q_{n}(\lambda E-A)P_{n}$ as

$\inf\left\\{{\sqrt{\big{\|}Q_{n}EP_{n}-\tilde{E}\big{\|}^{2}_{F}+\big{\|}Q_{n}AP_{n}-\tilde{A}\big{\|}^{2}_{F}}:\lambda\tilde{E}-\tilde{A}\text{
singular }}\right\\},$ (19)

as in [14].

###### Proposition 21

Consider a pencil $\lambda E-A$ of the form (8). If there exists a sequence of
unit norm vectors $\\{x_{n}\\}_{n=1}^{\infty}$ contained in the domain of $A$
and such that $Ax_{n}\to 0$, $Ex_{n}\to 0$ for $n\to\infty$, then for any
sequence of projections $P_{n},Q_{n}$ satisfying (17)–(18), the distance to
singularity (19) of the matrix pencils $Q_{n}(\lambda E-A)P_{n}$ converges to
zero with $n\to\infty$.

Proof. Let $\varepsilon>0$ be arbitrarily small and let $n$ be such that
$\left\|Ax_{n}\right\|<\varepsilon$, $\left\|Ex_{n}\right\|<\varepsilon$.
Moreover, let $m_{0}$ be such that for all $m>m_{0}$ one has
$\left\|P_{m}x_{n}\right\|>1-\varepsilon$,
$\left\|Q_{m}EP_{m}x_{n}\right\|<2\varepsilon$,
$\left\|Q_{m}AP_{m}x_{n}\right\|<2\varepsilon$. The existence of such an
$m_{0}$ follows by (17) and (18). Then with
$g_{m}=\frac{P_{m}x_{n}}{\left\|P_{m}x_{n}\right\|}$ for $m>m_{0}$, one has

$\sigma_{\min}^{2}\left(\left[\begin{array}[]{c}Q_{m}AP_{m}\\\
Q_{m}EP_{m}\end{array}\right]\right)\leq\left\|Q_{m}AP_{m}g_{m}\right\|^{2}+\left\|Q_{m}EP_{m}g_{m}\right\|^{2}<\frac{8\varepsilon^{2}}{(1-\varepsilon)^{2}}.$

As $\varepsilon>0$ was arbitrary, the left hand side has to converge to zero
with $m\to\infty$, which by an estimate in [14] implies that (19) converges to
0 with $n\to\infty$.

###### Remark 22

Let us make tow remarks. First, the assumptions of Proposition 21 almost
trivially imply that for any $\varepsilon>0$ there exist $A_{1}$, $E_{1}$,
both of norm bounded by $\varepsilon$, such that $\lambda(E+E_{1})-(A+A_{1})$
is a point singular pencil. Proposition 21 presents a stronger statement.
Second, convergence of (19) to 0 is true for _all_ $P_{n}$, $Q_{n}$, and thus
it may serve as a kind of an infinite dimensional singularity indicator.
However, it is rather easy to construct pencils that are clearly not singular
for which convergence of (19) to 0 is true for _some_ $P_{n}$, $Q_{n}$, see
Example 23 below.

###### Example 23

Let $T$ be the bilateral shift in $\ell^{2}(\mathbb{Z})$, i.e., $T$ acts on
the canonical basis as $Te_{j}=e_{j+1}$ for $j\in\mathbb{Z}$. As $T$ is
unitary, for the pencil $P(\lambda)=\lambda T-T$ we have
$s\big{(}P(\lambda)\big{)}=\\{1\\}$ and (11) does not hold and so the pencil
should not be seen as singular in any sense. However, the projections
$P_{n}=Q_{n}$ onto
$\operatorname{span}\\{e_{n-1},\dots,e_{0},e_{1},\dots,e_{n}\\}$ satisfy
(17)–(18) and the pencil $P_{n}(\lambda T-T)P_{n}$ is clearly singular as
$e_{n}\in\ker P_{n}TP_{n}$ for $n\geq 1$.

###### Remark 24

In some publications there is a debate whether an operator pencil as in
Example 20 should be considered as singular. This results from the fact that
the corresponding linear relation

$\\{(Ex,Ax):x\in\mathcal{X}\\}\subseteq\mathcal{X}\times\mathcal{X}$ (20)

is the identity relation. It was shown in [9, 8] that in finite dimensional
spaces there is a one-to-one correspondence between linear relations and
linear pencils that keeps the spectral properties. In particular, if
$\mathcal{X}=\mathcal{Y}$ is finite dimensional, then any linear pencil
$\lambda E-A$ with the corresponding relation (20) being equal to the identity
is a regular pencil, congruent to $\lambda I-I$. Apparently, in the infinite
dimensional setting the correspondence is more complex. See [25, 28] for
further work on regular and singular linear relations in infinite dimensional
spaces.

###  3.6 Approximative singularity

In this subsection we define and analyze the concept of approximate
singularity for operator pencils.

###### Definition 25

A sequence of $\mathcal{X}$-valued polynomials
$\\{p_{n}(\lambda)\\}_{n=1}^{\infty}$ is called a _right approximative
polynomial sequence for the operator pencil $\lambda E-A$_ of the form (8) if

$p_{n}(\lambda_{0})\neq 0,\quad\lim_{n\to\infty}\|p_{n}(\lambda_{0})\|\neq
0,\quad\operatorname{rev}p_{n}(\lambda_{0})\neq
0,\quad\lim_{n\to\infty}\|\operatorname{rev}p_{n}(\lambda_{0})\|\neq 0,$ (21)

while
$p_{n}(\lambda_{0})\in\mathcal{D}\left(E\right)\cap\mathcal{D}\left(A\right)$
for $n\geq 1$ and

$\lim_{n\to\infty}\left\|(\lambda_{0}E-A)p_{n}(\lambda_{0})\right\|=0,\
\lim_{n\to\infty}\left\|(\lambda_{0}A-E)\operatorname{rev}p_{n}(\lambda_{0})\right\|=0,$

for all $\lambda_{0}\in\mathbb{C}$. We say that a sequence
$\\{q_{n}(\lambda)\\}_{n=1}^{\infty}$ of $\mathcal{Y}$-valued polynomials is a
_left approximative polynomial sequence for the pencil $\lambda E-A$_ if it is
a right approximative polynomial sequence for the adjoint pencil $(\lambda
E^{*}-A^{*})$. Finally, we say that a pencil is _approximately singular_ if it
has either a left or right approximative polynomial sequence.

One may wonder if $p_{n}(\lambda_{0})\not\to 0$ for $\lambda_{0}\in\mathbb{C}$
implies $\operatorname{rev}p_{n}(\lambda_{0})\not\to 0$ for
$\lambda_{0}\in\mathbb{C}$, or, in other words, if the conditions on the
reversal in Definition 25 are necessary. The following example demonstrates
the necessity.

###### Example 26

Let $p_{n}(\lambda)=e_{1}+(\lambda^{n}/n!)e_{2}$ be a sequence of
$\mathbb{C}^{2}$-valued polynomials. Clearly $p_{n}(\lambda_{0})$ does not
converge to 0 for any $\lambda_{0}\in\mathbb{C}$, while
$\operatorname{rev}p_{n}(\lambda_{0})=\lambda^{n}e_{1}+e_{2}/n!$ does converge
to 0 for $|\lambda_{0}|<1$.

It is obvious that the situation of having a joint approximate kernel as in
Subsection 3.5 is covered by the definition of approximate singularity, but
(even in the finite dimensional case) the converse is not true. We have the
following lemma.

###### Lemma 27

Consider an operator pencil $P(\lambda)=\lambda E-A$ of the form (8).

  1. (i)

If $P(\lambda)$ has a right approximative polynomial sequence, then there
exist sequences of unit norm vectors
$\\{x_{n}\\}_{n}\subseteq\mathcal{D}\left(A\right)$ with $Ax_{n}\to 0$ and
$\\{y_{n}\\}_{n}\subseteq\mathcal{D}\left(E\right)$, with $Ey_{n}\to 0$, for
$n\to\infty$.

  2. (ii)

If $P(\lambda)$ has a sequence of unit norm vectors $\\{x_{n}\\}_{n}$ for
which $Ax_{n}\to 0$ and $Ex_{n}\to 0$, then it has a right approximative
polynomial sequence.

Proof. For the proof of (i) set $x_{n}=\frac{p_{n}(0)}{\|p_{n}(0)\|}$,
$y_{n}=\frac{\operatorname{rev}p_{n}(0)}{\|\operatorname{rev}p_{n}(0)\|}$,
$n>1$. For the proof of (ii) take constant polynomials
$p_{n}(\lambda)=x_{n}=\operatorname{rev}p_{n}(\lambda)$.

In Section 5 we present the class of dissipative Hamiltonian operator pencils
for which the condition in Lemma 27(ii) is already equivalent to being
approximatively singular.

Next we show that, among other properties, for the concept of approximative
singularity the criteria 1)–5) from Subsection 3.1 are satisfied. Let us start
with basic properties, that contain 1).

###### Proposition 28

Consider an operator polynomial $P(\lambda)=\lambda E-A$ of the form (8), with
$E\in\mathbf{B}(\mathcal{X},\mathcal{Y})$ and $A$ being closed and densely
defined operator from $\mathcal{D}\left(A\right)\subseteq\mathcal{X}$ to
$\mathcal{Y}$. Then the following statements hold:

  1. (i)

If $P(\lambda)$ has a right polynomial sequence then it has a right
approximative polynomial sequence.

  2. (ii)

If the spaces $\mathcal{X}$ and $\mathcal{Y}$ are finite dimensional then a
pencil is approximately singular if and only if it is singular.

Proof. Item (i) follows directly from Lemma 12. To see the statement in (ii)
observe that if $\lambda E-A$ has an approximative polynomial sequence, it is
not invertible for any $\lambda\in\mathbb{C}$. Hence, if $\mathcal{X}$ and
$\mathcal{Y}$ are finite dimensional, the pencil is singular.

Let us now move to criterion 2) from Subsection 3.1. We state the following
proposition for completeness of the presentation, the proof is obvious.

###### Proposition 29

Consider an operator pencil $P(\lambda)=\lambda E-A$ of the form (8), with
$E\in\mathbf{B}(\mathcal{X},\mathcal{Y})$ and $A$ being a closed and densely
defined operator from $\mathcal{D}\left(A\right)\subseteq\mathcal{X}$ to
$\mathcal{Y}$. Then the following statements hold:

  1. (i)

If $\lambda E-A$ has right approximative polynomial sequence, then the set of
approximate singularities coincides with the complex plane and infinity is an
approximate singularity.

  2. (ii)

If $\lambda E-A$ has left approximative polynomial sequence, then the set of
all singularities of $\lambda E-A$ coincides with the whole complex plane and
infinity is an approximate singularity.

We discuss now the remaining criteria of Subsection 3.1. Criterion 3) follows
directly from the definition. Criterion 4) follows from the fact that
$x_{n}\oplus y_{n}$ converges to zero if and only if both $x_{n}$ and $y_{n}$
converge to zero. Criterion 5) is obvious.

###### Example 30

The operator pencil $\lambda E-A$ of Example 10 has $\mathbb{C}$ as set of
approximate singularities and infinity is an approximate singularity as well.
(In fact, these are already point singularities.) However, the pencil is not
approximately singular according to Definition 25. Indeed, this would
contradict Criterion 4), as the pencil can be decomposed into two pencils with
nonempty sets of regular points.

In the following we present further examples. In order to check condition
(21), a useful tool is provided by the following proposition.

###### Proposition 31

For $p_{n}(\lambda)=\sum_{j=0}^{k_{n}}\lambda^{j}x^{(n)}_{j}$ with
$x^{(n)}_{j}\in\mathcal{X}$, let

$\xi:=\inf_{n\geq 0}\lambda_{\min}(\Xi^{(n)}),\quad\mbox{where
}\;\Xi^{(n)}:=\left[\left<x_{i}^{(n)},x_{j}^{(n)}\right>\right]_{i,j=0}^{k_{n}}\in{\mathbb{C}}^{k_{n}+1,k_{n}+1},\quad
n>0.$

If $\xi>0$ then neither $p_{n}(\lambda_{0})$ nor
$\operatorname{rev}p_{n}(\lambda_{0})$ converge to 0 for any
$\lambda_{0}\in\mathbb{C}$.

Proof. For $\lambda_{0}\in\mathbb{C}$, with
$u^{(n)}=\left[\begin{array}[]{ccc}\lambda_{0}^{0}\cdots&\lambda_{0}^{k_{n}}\end{array}\right]^{\top}\in{\mathbb{C}}^{k_{n}+1}$,
we have

$\displaystyle\big{\|}p_{n}(\lambda_{0})^{2}\big{\|}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i,j=0}^{k_{n}}\overline{\lambda_{0}}^{i}\lambda_{0}^{j}\left<x_{i}^{(n)},x_{j}^{(n)}\right>$
$\displaystyle=$ $\displaystyle u^{(n)*}\Xi^{(n)}u^{(n)}$ $\displaystyle\geq$
$\displaystyle\big{\|}u^{(n)}\big{\|}^{2}\lambda_{\min}(\Xi^{(n)})$
$\displaystyle\geq$ $\displaystyle\xi,\quad n\geq 0.$

Similarly, we obtain

$\displaystyle\big{\|}\operatorname{rev}p_{n}(\lambda_{0})\big{\|}^{2}$
$\displaystyle=$ $\displaystyle
u^{(n)*}\left[\left<x_{k_{n}-i}^{(n)},x_{k_{n}-j}^{(n)}\right>\right]_{ij=0}^{k_{n}}u^{(n)}$
$\displaystyle\geq$
$\displaystyle\big{\|}u^{(n)}\big{\|}^{2}\lambda_{\min}(S\,\Xi^{(n)}S)$
$\displaystyle=$
$\displaystyle\big{\|}u^{(n)}\big{\|}^{2}\lambda_{\min}(\Xi^{(n)})$
$\displaystyle\geq$ $\displaystyle\xi,\quad n\geq 0,$

where $S$ denotes the permutation matrix that reverses the order of indices.

Note that the condition provided in Proposition 31 is sufficient, but not
necessary.

###### Example 32

The (constant) sequence of $\mathbb{C}^{3}$-valued polynomials

$p_{n}(\lambda)=p(\lambda)=e_{1}+(\lambda+\lambda^{2})e_{2}+\lambda^{3}e_{3}$

clearly satisfies $\lim_{n\to\infty}p_{n}(\lambda_{0})\neq 0$ and
$\lim_{n\to\infty}\operatorname{rev}p_{n}(\lambda_{0})\neq 0$ for all
$\lambda_{0}\in\mathbb{C}$, but for all $n>0$ the matrix $\Xi^{(n)}$ is
singular.

###### Example 33

Let $\mathcal{X}=\bigoplus_{n=1}^{\infty}\mathcal{X}_{n}$, where
$\mathcal{X}_{n}=\mathbb{C}^{2n+1,2n+1}$ and define the operator pencil
$P(\lambda)=\lambda E-A$ with

$E=\bigoplus_{n=1}^{\infty}E_{n},\;A=\bigoplus_{n=1}^{\infty}A_{n},\quad
E_{n}=\left[\begin{array}[]{ccc}0&0&I_{n}\\\ 0&\alpha_{n}&0\\\
I_{n}&0&0\end{array}\right],\
A_{n}=\left[\begin{array}[]{ccc}\alpha_{n}&0&0\\\ 0&0&I_{n}\\\
0&I_{n}&0\end{array}\right],$

where $\alpha_{n}$ ($n\geq 1$) is a family of nonnegative parameters. First
note that if $\alpha_{n}=0$ for some $n$ then the pencil is point singular,
while if $\inf_{n\geq 1}\alpha_{n}>0$ then the pencil is not approximately
singular due to Lemma 27. Hence, the interesting case to consider is when
$\lim_{n\to\infty}\alpha_{n}=0$ but $\alpha_{n}\neq 0$ for all $n$.

Let $e_{j}^{(i)}$ denote the vector from $\mathcal{X}$ that has zero
components in all $\mathcal{X}_{k}$ different from $i$ and corresponds to the
$j$-th standard basis vector of $\mathbb{C}^{2i+1}$ in the component
$\mathcal{X}_{i}$. Then

$p_{n}(\lambda):=\sum_{j=0}^{n}\lambda^{j}e_{j+1}^{(n)},\quad n\geq 1$

is a sequence of polynomials, satisfying the assumptions of Proposition 31.
Indeed, we have $\Xi^{(n)}=I_{n}$ and hence $\xi=1$. Furthermore, we have

$Ae_{1}^{(n)}=\alpha_{n}e_{1}^{(n)},\quad
Ae_{j+1}^{(n)}=e_{n+j+1}^{(n)}=Ee_{j}^{(n)},\;j=1,\dots,n,\quad
Ee_{n+1}^{(n)}=\alpha_{n}e_{n+1}^{(n)}.$

Hence, for arbitrary $\lambda_{0}\in\mathbb{C}$ we have

$\displaystyle\left\|(\lambda E-A)p_{n}(\lambda_{0})\right\|$ $\displaystyle=$
$\displaystyle\|Ae_{1}^{(n)}\|+\sum_{j=1}^{n}|\lambda_{0}|^{j}\|Ee_{j+1}^{(n)}-Ae_{j}^{(n)}\|+|\lambda_{0}|^{n+1}\|Ee_{n+1}^{(n)}\|$
$\displaystyle=$
$\displaystyle\alpha_{n}(1+|\lambda_{0}|^{n+1})=\left\|(\lambda
A-E)\operatorname{rev}p_{n}(\lambda_{0})\right\|.$

Hence, for $\alpha_{n}=\frac{1}{(n+1)!}$ the pencil $\lambda E-A$ is
approximately singular.

Example 33 provides, in particular, an instance where the sequence of degrees
of any right approximative polynomial sequence is not bounded. However, in
general, right approximative polynomial sequences may contain right
approximative polynomial sequences whose individual polynomials have smaller
degrees than the original ones. Compared to the concept of singular chains
this corresponds to the observation that linearly dependent singular chains
contain singular chains of smaller length.

###### Example 34

Let $\mathcal{X}$ be the Hilbert space from Example 33 and let
$P(\lambda)=\lambda E-A$ with

$E=\bigoplus_{n=1}^{\infty}E_{n},\;A=\bigoplus_{n=1}^{\infty}A_{n},\quad
E_{n}=\frac{1}{n}\left[\begin{array}[]{ccc}1&0&0\\\ 0&0&I_{n}\\\
0&I_{n}&0\end{array}\right],\;A_{n}=\frac{1}{n}\left[\begin{array}[]{ccc}0&0&I_{n}\\\
0&1&0\\\ I_{n}&0&0\end{array}\right].$

Then as in Example 33 the polynomials

$p_{n}(\lambda):=\sum_{j=0}^{n}\lambda^{j}e_{j+1}^{(n)},\quad n\geq 1$

form a right approximate polynomial sequence for $\lambda E-A$. We also have
$Ee_{1}^{(n)},Ae_{1}^{(n)}\to 0$ for $n\to\infty$, so also the constant
polynomials $e_{1}^{(n)}$, $n\geq 1$, form a right approximate polynomial
sequence for $\lambda E-A$.

##  4 Operator differential-algebraic equations. Nonuniqueness of solutions.

In this section, we consider the effect of singularity of operator pencils on
the solutions of Cauchy problem associated with the linear ODAE (1).

$E\dot{x}(t)=Ax(t),\quad x(0)=x_{0},$ (22)

where $E\in\mathbf{B}(\mathcal{X},\mathcal{Y})$ and $A$ is a closed and
densely defined operator from $\mathcal{D}\left(A\right)\subseteq\mathcal{X}$
to $\mathcal{Y}$. Here and everywhere below $t$ is a real variable and
$\dot{x}$ denotes the partial derivative with respect to $t$. We present the
notions of classical solution and mild solutions to (22).

###### Definition 35

A continuous function $x:[0,T)\to\mathcal{X}$ is called a _classical solution_
of (22), if $x$ is continuously differentiable on $[0,T]$,
$x(t)\in\mathcal{D}\left(A\right)$ for each $t\in[0,T]$ and (22) holds.
Furthermore, $x(t)$ is called a _mild solution_ of (22), if it is continuous,
$x(0)=x_{0}$ and for all $t\in[0,T]$ we have
$\int_{0}^{t}x(s)ds\in\mathcal{D}\left(A\right)$, and
$Ex(t)-A\int_{0}^{t}x(s)ds=Ex_{0}.$

Obviously, a classical solution of (2) is also a mild solution of (2). We
refer the reader to [55] for a wide discussion on consistent initial values
and solvability criteria. However, note that the results of [55], as well as
many other reults on the subject, require nonemptyness of the set of regular
points. In the following we drop this assumption to study singularity of the
operator pencil and its relation with (non)uniqueness of the solutions of
(22).

Recall that if $E,A$ are matrices, then the analysis is well understood. If
the pencil $\lambda E-A$ is regular, then equation (22) has a solution for any
consistent initial value, see Theorem 2.12 of [32]. The situation is clearly
much more complicated in the operator case, where only sufficient conditions
for solvability are known, see e.g. [55, 45] for the case when the resolvent
set is nonempty. We will not discuss this topic here. Another classical fact
in the finite dimensional case is the non-existence of solutions of the
inhomogeneous DAE $E\dot{x}=Ax+g$ due to the presence of left singular blocks,
see Theorem 2.14(ii) of [32].

It follows from Theorem 2.14 in [32] that the matrix pencil $\lambda E-A$ has
a right singular polynomial if and only if the equation (22) with the initial
condition $x(0)=0$ has a nonzero solution. (Observe that existence of such
solutions is equivalent to nonuniqueness of solutions of (22) with nonzero
initial condition $x(0)=x_{0}$, if only such solutions exist.)

We will discuss an infinite dimensional analogue of this fact in the context
of the singularity concepts discussed so far in the paper. We begin with a
general result, showing non-uniqueness of classical solutions (hence, also
mild solutions) under the assumption of existence of a generalized singular
chain.

###### Theorem 36

Consider an operator pencil of the form (8) and the associated Cauchy problem
(22). Let $\\{a_{k}\\}_{k=1}^{\infty}$ with
$a_{k}\in\mathcal{D}\left(A\right)$ $(k\geq 1)$ be a nonzero sequence such
that

$Ea_{1}=0,\quad Ea_{k+1}=Aa_{k},\quad k\geq 1,$ (23)

and that $\left\|a_{k}\right\|$, $\left\|Aa_{k}\right\|$,
$\left\|Ea_{k}\right\|$ are bounded by $\left(\frac{k}{c}\right)^{k}$ for $k$
sufficiently large and some $c>0$. Then there exists a nonzero
$\mathcal{X}$-valued analytic function $f(t)$, defined for $|t|<1/(ce)$ and
satisfying

$f(t)\in\mathcal{D}\left(A\right),\quad E\dot{f}(t)=Af(t),\quad f(0)=0.$

Proof. Define $f(t)$ by the following series

$f(t)=\sum_{j=1}^{\infty}\frac{a_{j}}{j!}t^{j}.$ (24)

We then observe that the series $\sum_{j=1}^{\infty}\frac{Xa_{j}}{j!}t^{j}$
($X\in\\{I_{\mathcal{X}},A\\}$) and
$\sum_{j=1}^{\infty}\frac{Xa_{j}}{(j-1)!}t^{j-1}$
($X\in\\{I_{\mathcal{X}},E\\}$) are convergent for $|t|<1/ce$ due to

$\limsup_{k\to\infty}\left(\frac{\left(\frac{k}{c}\right)^{k}}{k!}\right)^{1/k}=\frac{1}{ce}>0.$

Hence, $f(t)$ is well defined, and by closedness of $A$ we have
$f(t)\in\mathcal{D}\left(A\right)$ for small $t$. By definition $f(0)$ is zero
and a direct calculation shows $E\dot{f}(t)=Af(t)$ everywhere where the series
defining $f(t)$ converges.

Based on this we obtain the relation between the singularity notions for
operator pencils and (non)uniqueness of the solutions of the corresponding
Cauchy problem.

###### Corollary 37

Consider the Cauchy problem (22). If the operator pencil $\lambda E-A$ has a
right singular polynomial, then there exists a nonzero $\mathcal{X}$-valued
polynomial $f(t)$ with $f(0)=0$ satisfying $f(t)\in\mathcal{D}\left(A\right)$,
and $E\dot{f}(t)=Af(t)$ for $t\in\mathbb{R}$.

Proof. Let $p(\lambda)=a_{k+1}+\cdots+\lambda^{k}a_{1}$ be a right singular
polynomial of $\lambda E-A$. We show that after appending the coefficients
with an infinite number of zeros, the sequence $\\{a_{k}\\}_{k=1}^{\infty}$
satisfies the assumptions of Theorem 36. By Lemma 12 we may assume that
$a_{k+1}\neq 0$, hence the sequence $\\{a_{k}\\}_{k=1}^{\infty}$ is nonzero.
Now we show that $a_{k}\in\mathcal{D}(A)$ for $k\geq 1$. By definition, for
every $\lambda_{0}\in\mathbb{C}$ the vector $p(\lambda_{0})$ is in
$\mathcal{D}\left(A\right)$. Taking distinct
$\lambda_{0},\dots,\lambda_{k}\in\mathbb{C}$ we obtain $k+1$ vectors

$b_{j}=\sum_{i=0}^{k}\lambda_{j}^{i}a_{i}\in\mathcal{D}\left(A\right),\quad
j=0,\dots k.$

Consider the Vandermonde matrix
$\Lambda:=\left[\begin{array}[]{c}\lambda_{j}^{i}\end{array}\right]_{ij=0}^{k+1}$
and its inverse
$\Lambda^{-1}=:\left[\begin{array}[]{c}\mu_{ij}\end{array}\right]_{ij=0}^{k+1}$.
Then

$a_{j}=\sum_{i=0}^{k}\mu_{ji}b_{i}\in\mathcal{D}\left(A\right),\quad j=0,\dots
k.$

Since $p(\lambda)$ is a right singular polynomial we have, by definition,
$(\lambda E-A)p(\lambda)=0$, which immediately gives (23). Now observe that
(24) is also a polynomial, hence $E\dot{p}=Ap$ for all $t\in\mathbb{R}$.

###### Example 38

Let $E$ be the closure of the forward shift operator and let $A$ be the
closure of a diagonal operator in $\ell^{2}$ defined by

$Ee_{1}=0,\ Ee_{k+1}=e_{k},\quad Ae_{k}=(k+1)e_{k},\quad(k\geq 1),$

extending them as usually to linear operators on the linear span of basis
vectors and taking the closure. Then $E$ is bounded but $A$ is not. The
sequence $a_{k}=k!e_{k}$ satisfies the assumptions of Theorem 36. On the other
hand $A$ is invertible and hence $\lambda E-A$ neither has left nor right
singular polynomials.

We also have a relation between non-uniquenss of solutions of (22) and the
existence of a right singular function for $\lambda E-A$.

###### Remark 39

If $\left\|a_{k}\right\|$, $\left\|Aa_{k}\right\|$, $\left\|Ea_{k}\right\|$
are bounded by $(k/c)^{-k}$ for $k$ sufficiently large and some $c>0$ then
$x(\lambda)=\sum_{j=1}^{\infty}a_{j}\lambda^{-j}$ is a right singular function
of $\lambda E-A$, as in Section 3.4. However, for $A,E$ as in Example 38 the
series $x(\lambda)=\sum_{j=1}^{\infty}a_{j}\lambda^{-j}$ diverges for all
$\lambda\neq 0$,

Finally, we discuss whether non-uniqueness of the solutions of (22) is
connected with approximate singularity of $\lambda E-A$. Apparently, there
does not seem to be any relation, see the following two examples.

###### Example 40

Let $E$ be the shift on the canonical basis $e_{1},e_{2},\dots$ of $\ell^{2}$,
i.e., $Ee_{k}=e_{k-1}$,$Ee_{1}=0$ and let $A=I_{\ell^{2}}$. Then $E$, $A$ and
$a_{k}=e_{k}$ ($k\geq 1)$ satisfy the setting of Proposition 36 and
$f(t)=\sum_{j=1}^{\infty}\frac{e_{j}}{j!}t^{j}$ is a solution of
$E\dot{f}=Af$, with $f(0)=0$. On the other hand $0$ is clearly a regular point
of the pencil $\lambda E-A$. In particular, the pencil is not approximately
singular.

Furthermore, there exist nontrivial solutions of (22) as e.g., with this $f$,
for each $\alpha\in\mathbb{R}$, the function $g$ satisfying $g(0)=e_{1}$ the
function $g(t)=e_{1}+\sum_{j=2}^{\infty}\frac{1}{(k-1)!}e_{k}t^{k-1}+\alpha
f(t)$ is a solution.

###### Example 41

Consider $A=E=\mbox{\rm diag}(1,1/2,1/3,...)$ in $\ell^{2}$ from Example 20.
The DAE $E\dot{x}=Ax$ has precisely one solution for any initial condition
$x(0)=x_{0}\in\ell^{2}$, namely $x(t)=e^{\operatorname{i}t}x_{0}$.

In this section we have studied the relationship between the different
singularity concepts and the uniqueness of solutions of the Cauchy problem
(22). In the next section we discuss the particular class of dissipative
operator pencils.

##  5 Dissipative Hamiltonian Operator pencils

As discussed in the introductory section a major motivation for the analysis
of operator DAEs is the study of dissipative Hamiltonian operator pencils. We
will consider these pencils in the slightly more general form

$P(\lambda)=\lambda E-BQ,$ (25)

where

  1. (i)

$E\in{\mathbf{B}}(\mathcal{X},\mathcal{Y})$,
$Q\in\mathbf{B}(\mathcal{X},\mathcal{Y})$ is invertible and $Q^{*}E$ is
selfadjoint and nonnegative , i.e., $\left<Q^{*}Ex,x\right>\geq 0$ for all
$x\in\mathcal{X}$;

  2. (ii)

$B$ is a closed, densely defined and dissipative operator in $\mathcal{Y}$,
i.e., $\mbox{\rm Re}\left<Bx,x\right>\leq 0$ for all
$x\in\mathcal{D}\left(B\right)\subseteq\mathcal{Y}$.

Such pencils satisfy the assumptions introduced in Section 2, as $BQ$ is
closed and densely defined. In particular, the set of regular points is open
(possibly empty). We note that this class is more general than the class
studied in [45], as we neither assume maximal dissipativity of $B$ nor that
$\ker E$ needs to be an invariant subspace for $Q$.

###  5.1 Singularity of dissipative Hamiltonian operator pencils

In [43] singularity of dissipative Hamiltonian pencils was investigated in the
finite-dimensional case. In particular, it was shown that the presence of
eigenvalues in the open right half plane of matrix pencils of the form (25)
already implies singularity of the pencil, which furthermore is equivalent to
all three matrices $E$, $J$ and $R$ having a common kernel.

In this section we extend these results to the infinite dimensional case. By
doing so we obtain an essentially simplified framework for studying
singularity, compared to Section 3.

In a recent paper [45] regularity and singularity of related pencils is
studied, though in a slighty different operator setting. The results of [45]
are, however, similar to Theorems 44, 45 and 46 below: regularity is
equivalent to the open right half-plane being contained in the set of regular
points.

###### Proposition 42

Consider an operator pencil of the form (25), let $\lambda_{0}\in\mathbb{C}$
with $\mbox{\rm Re}\lambda_{0}>0$ and let vectors $x_{n}\in\mathcal{D}(BQ)$,
$n\in\mathbb{N}$, form a bounded sequence. Then for $n\to\infty$, we have

$P(\lambda_{0})x_{n}\to 0\quad\mbox{ if and only if }\quad Ex_{n}\to 0\mbox{
and }BQx_{n}\to 0.$

In particular, we have

$\ker P(\lambda_{0})=\ker E\cap\ker(BQ).$ (26)

If, additionally, $\mathcal{X}=\mathcal{Y}$, $Q=I_{\mathcal{X}}$ and $B=J-R$,
with both $J$ and $R$ being closed,
$\mathcal{D}\left(J\right)\subseteq\mathcal{D}\left(R\right)$ or
$\mathcal{D}\left(R\right)\subseteq\mathcal{D}\left(J\right)$ and
$\left<Jx,x\right>\in\operatorname{i}\mathbb{R}$, $\left<Rx,x\right>\geq 0$
for $x\in\mathcal{D}\left(R\right)\cap\mathcal{D}\left(J\right)$ then

$\ker P(\lambda_{0})=\ker E\cap\ker J\cap\ker R=\ker(E^{2}+R^{2}-J^{2}).$ (27)

Proof. Let $P(\lambda_{0})x_{n}\to 0$, i.e., $(\lambda_{0}E-BQ)x_{n}\to 0$.
Then, $(\lambda_{0}Q^{*}E-Q^{*}BQ)x_{n}\to 0$ and
$\left<(\lambda_{0}Q^{*}E-Q^{*}BQ)x_{n},x_{n}\right>\to 0$. Taking the real
part and using that $\left<\big{(}\mbox{\rm
Re}(\lambda_{0})Q^{*}E\big{)}x_{n},x_{n}\right>\geq 0$ as well as $\mbox{\rm
Re}\left<(Q^{*}BQ)x_{n},x_{n}\right>=\mbox{\rm
Re}\left<BQx_{n},Qx_{n}\right>\leq 0$, we obtain
$\left<Q^{*}Ex_{n},x_{n}\right>\to 0$. The Cauchy-Schwarz inequality for the
semidefinite inner product $\left<Q^{*}E\cdot,\cdot\right>$ and the
boundedness of $Q^{*}E$ then yield that

$\left<Q^{*}Ex_{n},Q^{*}Ex_{n}\right>\leq\left<Q^{*}Ex_{n},x_{n}\right>^{1/2}\left<(Q^{*}E)^{2}x_{n},Q^{*}Ex_{n}\right>^{1/2}\leq\left<Q^{*}Ex_{n},x_{n}\right>^{1/2}\cdot\left\|Q^{*}E\right\|^{3/2}\cdot\|x_{n}\|.$
(28)

Hence, $Q^{*}Ex_{n}\to 0$ and thus $Ex_{n}\to 0$, since $Q$ is invertible, and
$BQx_{n}\to 0$. The converse implication is trivial and to see (26) take
$x\in\ker P(\lambda_{0})$ and a constant sequence $x_{n}=x$.

To prove the second part of the claim, let $x\in\ker P(\lambda_{0})$. Due to
the first part we obtain $Ex=0$ and $(J-R)x=0$. Since $\left<Jx,x\right>$ is
purely imaginary and $\left<Rx,x\right>$ is real, we have
$\left<Rx,x\right>=0$. Recall that $R$ is positive semidefninite (not
necessarily bounded), hence,
$|\left<Rx,y\right>|\leq\left<Rx,x\right>^{1/2}\left<Ry,y\right>^{1/2}=0$ for
any $y\in\mathcal{D}\left(R\right)$. Consequently $x\in\ker R$ and finally
$Jx=0$ as well, which shows that $x\in\ker E\cap\ker R\cap\ker J$. The
inclusion $\ker E\cap\ker R\cap\ker J\subseteq\ker(E^{2}-J^{2}+R^{2})$ is
trivial. If now $x\in\ker(E^{2}-J^{2}+R^{2})$ then
$\left<(E^{2}-J^{2}+R^{2})x,x\right>=0$ and due to the fact that $E^{2},R^{2}$
and $-J^{2}$ are positive semidefinite we have $Ex=Rx=Jx=0$, which shows
$x\in\ker P(\lambda_{0})$. This finishes the proof of (27).

###### Remark 43

If we are in the special case that $Q=I_{\mathcal{X}}$ in Proposition 42 and
if the coefficient $R$ of the pencil $P(\lambda)$ is not only symmetric, but
also bounded (and hence self-adjoint), then the condition
$QBx_{n}=(J-R)x_{n}\to 0$ is in fact equivalent to $Jx_{n},Rx_{n}\to 0$.
Indeed, $(J-R)x_{n}\to 0$ implies $\langle Jx_{n},x_{n}\rangle$,$\langle
Rx_{n},x_{n}\rangle\to 0$ and from the latter we obtain $Rx_{n}\to 0$ by the
same reasoning as in (28). But then we must also have $Jx_{n}\to 0$. Without
the boundedness of $R$ one can show weak convergenve of $Rx_{n}$ to $0$, i.e.,
$\left<Rx_{n},y\right>\to 0$ for all $y\in\mathcal{X}$.

The next results investigates the effects of the presence of singular points
in the open right half plane. We will distinguish between point singularities,
approximative singularities and general singular points. Let us point out the
essential difference with the general case discussed in Section 3. From
Theorem 44 it follows that $s_{p}\big{(}P(\lambda)\big{)}=\mathbb{C}$ _is_
equivalent to being point singular (cf. in contrast Example 14), and from
Theorem 45 it follows that $s_{ap}\big{(}P(\lambda)\big{)}=\mathbb{C}$ _is_
equivalent to being approximately singular, cf. Example 30).

###### Theorem 44

Consider an operator pencil $P(\lambda)$ of the form (25). Then the following
conditions for point singularities are equivalent:

  1. (a)

$s_{\operatorname{p}}\big{(}P(\lambda)\big{)}=\mathbb{C}$ and $\infty$ is a
point singularity;

  2. (b)

$s_{\operatorname{p}}\big{(}P(\lambda)\big{)}\cap\left\\{\lambda_{0}\in{\mathbb{C}}:\mbox{\rm
Re}\lambda_{0}>0\right\\}\neq\emptyset$;

  3. (c)

$\ker E\cap\ker(BQ)\neq\left\\{0\right\\}$;

  4. (d)

the pencil $P(\lambda)$ has a right singular polynomial.

Proof. The implication (a) $\Rightarrow$ (b) is trivial, (b) $\Rightarrow$ (c)
follows from Proposition 42. To see (c) $\Rightarrow$ (d) let $p(\lambda)=x$,
where $x$ is a nonzero element of $\ker E\cap\ker BQ$. Clearly it is a right
singular polynomial, see Definition 11. The implication (d) $\Rightarrow$ (a)
is a special instance of Proposition 13.

For approximate singularities we have the following equivalence result.

###### Theorem 45

Consider a pencil $P(\lambda)$ of the form (25). Then the following conditions
for approximative singularities are equivalent:

  1. (a)

$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}=\mathbb{C}$ and $\infty$ is an
approximate singularity;

  2. (b)

$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}\cap\left\\{\lambda_{0}\in{\mathbb{C}}:\mbox{\rm
Re}\lambda_{0}>0\right\\}\neq\emptyset$;

  3. (c)

there exists a sequence of unit norm vectors $x_{n}\in\mathcal{D}(BQ)$ such
that $Ex_{n},BQx_{n}\to 0$;

  4. (d)

the pencil $P(\lambda)$ has a right approximate singular polynomial sequence.

Proof. The implication (a) $\Rightarrow$ (b) is trivial, (b) $\Rightarrow$ (c)
follows from Proposition 42. The implication (c) $\Rightarrow$ (d) follows
from the definition and (d) $\Rightarrow$ (a) is a special case of Theorem 29.

Finally, we present a result on the complete set of singular points.

###### Theorem 46

Consider a pencil $P(\lambda)$ of the form (25). Then the following conditions
are equivalent:

  1. (a)

$s\big{(}P(\lambda)\big{)}=\mathbb{C}$ and $\infty$ is a singularity;

  2. (b)

$s\big{(}P(\lambda)\big{)}\cap\left\\{\lambda_{0}\in{\mathbb{C}}:\mbox{\rm
Re}\lambda_{0}>0\right\\}\neq\emptyset$.

Proof. The implication $(a)\Rightarrow(b)$ is trivial. Let us assume that
$(a)$ does not hold. In particular, the set of approximate singularities is
not the whole complex plane. By Theorem 45, we have that
$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}\cap\left\\{\lambda_{0}\in{\mathbb{C}}:\mbox{\rm
Re}\lambda_{0}>0\right\\}=\emptyset$. But then it follows immediately from
Proposition 5(ii) that (b) does not hold.

###### Remark 47

We have shown that conditions (10), (11) and (12) are not equivalent for
general operator pencils. In fact, they are also not equivalent for operator
pencils of the form (25). Suitable counterexamples are essentially the same
ones as before. For the first situation take any pencil $\lambda E-E$ with $E$
selfadjoint bounded, positive semidefinite, with zero in the approximative
spectrum but not in the point spectrum. For the second one take the pencil in
(13) with $\tilde{A}=\operatorname{i}B$, where $B$ is symmetric but not
selfadjoint.

Using Theorem 21 and Theorem 45 we have the following corollary.

###### Corollary 48

If $\lambda E-BQ$ is a pencil as in (25) satisfying one of the equivalent
conditions in Theorem 45 then for any sequence of projections $P_{n}$, $n\geq
1$ satisfying (17)–(18) the distance to singularity of the finite dimensional
operator pencils $P_{n}(\lambda E-A)P_{n}$ converges to zero for $n\to\infty$.

###  5.2 Uniqueness of solutions of the corresponding operator DAEs

Consider the dissipative operator DAE

$E\dot{x}(t)=BQx(t),\quad x(0)=x_{0},\quad t\in[0,T],$ (29)

with $E,B,Q$ as in (25). We refer to Section 4 for the definition of classical
and mild solutions. It was also presented there that neither nonemptyness of
the set of regular points of $\lambda E-A$ implies uniqueness of the solutions
of (29) nor conversely, see Example 40 and Example 41, respectively. While
Example 41 is already of the dissipative Hamiltonian form (25), here
$Q=I_{\ell^{2}}$, Example 40 cannot be written in this form. This will become
apparent, as we will show that for dissipative Hamiltonian pencils with the
set of point singularities not equal to the whole complex plane the solution
of (29) is unique (if it exists). This is a typical fact in energy based
approaches, see, e.g., [22]. The key result that is needed for this is the
power balance equation, see [46] for a detailed study,

$\left<Ef(t_{0}),Qf(t_{0})\right>-\left<Ef(0),Qf(0)\right>=2\mbox{\rm
Re}\int_{0}^{t_{0}}\left<Q^{*}BQf(t),f(t)\right>dt,\quad{t_{0}\in[0,T]},$ (30)

which holds for any classical solution of (29). Indeed, if $f(t)$ is a
classical solution, then

$\frac{d}{dt}\left<Ef(t),Qf(t)\right>=2\mbox{\rm
Re}\left<E\dot{f}(t),Qf(t)\right>=2\mbox{\rm
Re}\left<Q^{*}BQf(t),f(t)\right>,$

which after integration gives (30). Note that Theorem 49 provides in fact
another equivalent condition to the conditions (a)–(d) from Theorem 44.
However, for presentation reasons, we work with a negated version of (c).

###### Theorem 49

Consider a linear operator pencil of the form (25). Then the following
statements are equivalent:

  1. (c’)

$\ker E\cap\ker BQ=\\{0\\}$;

  2. (e’)

if $x_{1}(t),x_{2}(t)$ are two mild solutions of (29) (with the same initial
condition), then $x_{1}(t)=x_{2}(t)$ for $t\in[0,T]$.

Proof. To show (e’)$\Rightarrow$(c’) assume that (e’) holds and that we have
$\ker E\cap\ker BQ\neq\\{0\\}$. Hence, there exists a nonzero continuously
differentiable function $x$ satisfying $x(0)=0$ and $x(t)\in\ker
E\cap\ker(BQ)$ for all $t\in[0,T]$. Assume that $x_{1}(t)$ is a mild solution
of (29). Then $x_{2}(t)=x_{1}(t)+x(t)$ is a mild solution of (22) as well.
Further $x_{2}(t)$ and $x_{1}(t)$ are not equal almost everywhere, which is a
contradiction.

To show the converse implication assume that $\ker E\cap\ker(BQ)=\\{0\\}$.
Suppose that $x_{1}(t),x_{2}(t)$ are two mild solution of (29) with the same
initial condition. Hence, $u(t)=x_{1}(t)-x_{2}(t)$ is a mild solution of (29)
with the initial condition $u(0)=0$. Therefore, $f(t)=\int_{0}^{t}u(s)ds$ is a
classical solution of (29) with the initial condition $f(0)=0$. Hence, (30)
holds, and for any $t_{0}\in[0,T]$ the left hand side is nonnegative while the
right hand side is nonpositive. Thus, they are both zero. In particular,
$\left<Q^{*}Ef(t_{0}),f(t_{0})\right>=0$, which gives $Ef(t_{0})=0$, since $Q$
is invertible. On the other hand, $2\mbox{\rm
Re}\int_{0}^{t_{0}}\left<Q^{*}BQf(t),f(t)\right>dt=0$. As $f(t)$ is continuous
and $\ker(BQ)=\ker(Q^{*}BQ)$ is a closed space, we have that $BQf(t_{0})=0$.
Hence, $f(t_{0})\in\ker E\cap\ker BQ=\\{0\\}$, by (c’). Consequently,
$x_{1}(t_{0})=x_{2}(t_{0})$ for $t_{0}\in[0,T]$.

###  5.3 Pencils of block operator matrices

Motivated by the linearized Navier-Stokes equations, in particular by (5), we
consider the following block operator matrices

$E=\left[\begin{array}[]{cc}M&0\\\ 0&0\end{array}\right],\quad
A=B=\left[\begin{array}[]{cc}D&-C^{*}\\\ C&0\end{array}\right],\quad
Q=I_{\mathcal{X}_{1}\oplus\mathcal{X}_{2}}$ (31)

where $\mathcal{X}=\mathcal{X}_{1}\oplus\mathcal{X}_{2}$,
$M\in{\mathbf{B}}(\mathcal{X}_{1})$ is positive and invertible, $D$ is a
closed, densely defined, nonnegative operator and $C$ is closed and densely
defined with $\mathcal{D}\left(D\right)\subseteq\mathcal{D}\left(C\right)$.
Clearly,
$\mathcal{D}\left(A\right)=\mathcal{D}\left(D\right)\oplus\mathcal{D}\left(C^{*}\right)$,

$\ker A\cap\ker E=\\{0\\}\oplus\ker C^{*},$ (32)

and

$C(D+\lambda M)^{-1}\in\mathbf{B}(\mathcal{X}_{1}).$ (33)

Note that the pencil $\lambda E-A$ has the form (25) with

$J=\left[\begin{array}[]{cc}0&-C^{*}\\\ C&0\end{array}\right],\quad
R=\left[\begin{array}[]{cc}D&0\\\ 0&0\end{array}\right].$

The theory developed in this paper is summarized in the following result.
Recall that $s_{p}\big{(}P(\lambda)\big{)}$ stands for the set of eigenvalues
of $P(\lambda)$, see Definition 4.

###### Theorem 50

Consider an operator pencil $P(\lambda)=\lambda E-A$ with $E,A$ as in (31).
Then the following are equivalent

  1. (i)

$\ker C^{*}=\\{0\\}$;

  2. (ii)

$s_{\operatorname{p}}\big{(}P(\lambda)\big{)}\neq{\mathbb{C}}$;

  3. (iii)

$s_{\operatorname{p}}\big{(}P(\lambda)\big{)}\subseteq\\{z\in\mathbb{C}:\mbox{\rm
Re}z\leq 0\\}$;

  4. (iv)

$\lambda E-A$ has no right singular polynomial;

  5. (v)

any two mild solutions of (29) (with the same initial condition) are equal for
$t\in[0,T]$.

Furthermore, $s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}\neq{\mathbb{C}}$
if and only if $\left\|C^{*}y\right\|\geq c\left\|y\right\|$ for all
$y\in\mathcal{D}\left(C^{*}\right)$ and some $c>0$.

Proof. Note that (i) is equivalent to $\ker E\cap\ker A=\\{0\\}$, due to
formula (32). The equivalence of $\ker E\cap\ker A=\\{0\\}$ with any of the
conditions (ii)–(v) follows from Theorem 44(a) and Theorem 49.

To see the ‘furthermore’ part first suppose that
$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}={\mathbb{C}}$. By Theorem 44(b)
there exists a sequence $\left\\{x_{n}\oplus y_{n}\right\\}_{n=1}^{\infty}$ of
unit vectors in $\mathcal{D}\left(A\right)$ such that $Mx_{n}\to 0$ and
$Dx_{n}\to 0$, $C^{*}y_{n}\to 0$ and $Cx_{n}\to 0$. Then as $Mx_{n}\to 0$ and
$M$ is invertible we have $x_{n}\to 0$, $\left\|y_{n}\right\|\to 1$. As
$C^{*}y_{n}\to 0$ we see that $C^{*}$ is not bounded from below.

Conversely, if $C^{*}$ is not bounded from below, then there exists a unit
norm sequence $y_{n}$ with $C^{*}y_{n}\to 0$. The sequence
$\\{z_{n}\\}_{n=1}^{\infty}$ with $z_{n}:=0\oplus y_{n}$ consists of unit norm
vectors, and we have $Az_{n}\to 0$, $Ez_{n}\to 0$, hence,
$s_{\operatorname{ap}}\big{(}P(\lambda)\big{)}={\mathbb{C}}$.

###### Remark 51

While a similar characterization of
$s\big{(}P(\lambda)\big{)}\neq{\mathbb{C}}$ seems to be impossible to obtain,
we present a simple sufficient condition. To this end, assume that $C$ is
invertible and $C^{-*}DC^{-1}$ is a densely defined and bounded operator (on
its domain), and denote by $T\in\mathbb{B}(\mathcal{X})$ the unique bounded
extension of $C^{-*}DC^{-1}$. Then the open right half plane is contained in
the resolvent set of $P(\lambda)$ and the resolvent is given by

$(\lambda E-A)^{-1}=\left[\begin{array}[]{cc}0&C^{-1}\\\
-(C^{*})^{-1}&-T+\lambda(C^{*})^{-1}MC^{-1}\end{array}\right]$

and satisfies the growth condition

$\left\|(\lambda E-A)^{-1}\right\|\leq m|\lambda|,\quad\mbox{\rm
Re}\lambda>0,$

with some $m\geq 0$. This estimate can be further used to obtain solvability
of the Cauchy problem (22) via the method of semigroups, see [55].

We are also able to analyse Example 3 in a similar abstract way. Consider the
operators

$E=\begin{bmatrix}Y&0&0\\\ 0&A_{0}&0\\\ 0&0&M\end{bmatrix},\quad
A=B=\begin{bmatrix}0&-A_{0}&D^{*}\\\ A_{0}&0&0\\\ -D&0&-K\end{bmatrix},\quad
Q=I_{\mathcal{X}_{1}\oplus\mathcal{X}_{1}\oplus\mathcal{X}_{2}},$ (34)

where
$\mathcal{X}=\mathcal{Y}=\mathcal{X}_{1}\oplus\mathcal{X}_{1}\oplus\mathcal{X}_{2}$,
$Y,A_{0}\in\mathbf{B}(\mathcal{X}_{1})$, $M\in\mathbf{B}(\mathcal{X}_{2})$ and
$Y,A_{0},M,K$ are positive semidefinite and
$\mathcal{D}(K)\subseteq\mathcal{D}(D^{*})$.

###### Theorem 52

Consider an operator pencil $P(\lambda)=\lambda E-A$ with $E,A$ as in (34).
Then the following are equivalent

  1. (i)

$\ker(Y^{2}+A_{0}^{2}+D^{*}D)\oplus\ker A_{0}\oplus(\ker
D^{*}\cap\ker(K^{2}+M^{2}))=\\{0\\}$;

  2. (ii)

$s_{\operatorname{p}}\big{(}P(\lambda)\big{)}\neq{\mathbb{C}}$;

  3. (iii)

$s_{\operatorname{p}}\big{(}P(\lambda)\big{)}\subseteq\\{z\in\mathbb{C}:\mbox{\rm
Re}z\leq 0\\}$;

  4. (iv)

$\lambda E-A$ has no right singular polynomial;

  5. (v)

any two mild solutions of (29) (with the same initial condition) are equal for
$t\in[0,T]$.

Proof. Note that (i) is equivalent to $\ker E\cap\ker A=\\{0\\}$. Indeed,
observe that

$E^{2}+R^{2}-J^{2}=\begin{bmatrix}Y^{2}+A_{0}^{2}+D^{*}D&0&0\\\
0&2A_{0}^{2}&-A_{0}D^{*}\\\ 0&-DA_{0}&DD^{*}+K^{2}+M^{2}\end{bmatrix}$

and

$\begin{bmatrix}2A_{0}^{2}&-A_{0}D^{*}\\\
-DA_{0}&DD^{*}+K^{2}+M^{2}\end{bmatrix}=\begin{bmatrix}A_{0}&0\\\
0&D\end{bmatrix}\begin{bmatrix}2I_{\mathcal{X}_{1}}&-I_{\mathcal{X}_{1}}\\\
-I_{\mathcal{X}_{1}}&I_{\mathcal{X}_{1}}\end{bmatrix}\begin{bmatrix}A_{0}&0\\\
0&D^{*}\end{bmatrix}+\begin{bmatrix}0&0\\\ 0&K^{2}+M^{2}\end{bmatrix},$

which is a sum of two positive-definite operators. We apply Proposition 42,
once to the sum above and once to original $E,A$ we obtain that

$\ker E\cap\ker
A=\ker(E^{2}+R^{2}-J^{2})=\ker(Y^{2}+A_{0}^{2}+D^{*}D)\oplus\ker
A_{0}\oplus(\ker D^{*}\cap\ker(K^{2}+M^{2})).$

Now the equivalence of $\ker E\cap\ker A=\\{0\\}$ with any of the conditions
(ii)–(v) follows from Theorem 44(a) and Theorem 49.

##  6 Conclusions

The spectral theory for operator pencils has been studied. Three different
concepts of singular operator pencils have been introduced and analyzed in
detail. Many examples illustrate the subtle differences. The results are then
applied to operator pencils arising in dissipative differential-algebraic
equations and it is shown that many results known from the finite dimensional
case of matrix pencils extend directly to the infinite dimensional setting.

## Acknowledgment

The authors are indebted to Jan Stochel for valuable ideas and discussions.

V.M. has been supported by Deutsche Forschungsgemeinschaft (DFG) through the
SPP1984 “Hybrid and multimodal energy systems” Project: _Distributed Dynamic
Security Control_ and by the DFG Research Center Math+, Project No. 390685689.
_Advanced Modeling, Simulation, and Optimization of Large Scale Multi-Energy
Systems_.

M.W. was supported by the Jagieloinan University ID.UJ grant, decision No.
PSP: U1U/P06/NO/02.12

## References

  * [1] F. Achleitner, A. Arnold, and V. Mehrmann.  Hypocoercivity and controllability in linear semi-dissipative ODEs and DAEs.  Z. Angew. Math. Mech., 103:e202100171, 2021.
  * [2] R. Altmann, V. Mehrmann, and B. Unger.  Port-Hamiltonian formulations of poroelastic network models.  Math. and Comp. Modelling Dynamical Systems, 27:429–452, 2021.
  * [3] R. Altmann and P. Schulze.  A port-Hamiltonian formulation of the Navier-Stokes equations for reactive flows.  Sys. Control Lett., 100:51–55, 2017.
  * [4] S. Aoues, F. L. Cardoso-Ribeiro, D. Matignon, and D. Alazard.  Modeling and control of a rotating flexible spacecraft: A port-Hamiltonian approach.  IEEE Trans. Contr. Sys. Techn., 27(1):355–362, 2017.
  * [5] A. Baaiu, F. Couenne, D. Eberard, C. Jallut, Y. Legorrec, L. Lefèvre, and B. Maschke.  Port-based modelling of mass transport phenomena.  Math. and Comp. Model. Dynam. Systems, 15(3):233–254, 2009.
  * [6] H. Bansal, P. Schulze, M. H. Abbasi, H. Zwart, L. Iapichino, W. H. A. Schilders, and N. Wouw.  Port-Hamiltonian formulation of two-phase flow models.  Sys. Control Lett., 149:104881, 2021.
  * [7] C. K. Batchelor and G. K. Batchelor.  An Introduction to Fluid Dynamics.  Cambridge University Press, 2000.
  * [8] T. Berger, H. de Snoo, C. Trunk, and H. Winkler.  A Jordan-like decomposition for linear relations in finite-dimensional spaces.  arXiv preprint arXiv:2209.14234, 2022.
  * [9] T. Berger, C. Trunk, and H. Winkler.  Linear relations and the Kronecker canonical form.  Linear Algebra Appl., 488:13–44, 2016.
  * [10] M. A. Biot.  General theory of three-dimensional consolidation.  J. Appl. Phys., 12(2):155–164, 1941.
  * [11] D. Boffi, F. Brezzi, and M. Fortin.  Mixed Finite Element Methods and Applications, volume 44.  Springer, 2013.
  * [12] S. Bögli.  Local convergence of spectra and pseudospectra.  Journal of Spectral Theory, 8(3):1051–1098, 2018.
  * [13] S. Bögli and M. Marletta.  Essential numerical ranges for linear operator pencils.  IMA Journal of Numerical Analysis, 40(4):2256–2308, 2020.
  * [14] R. Byers, C. He, and V. Mehrmann.  Where is the nearest non-regular pencil.  Linear Algebra Appl., 285:81–105, 1998.
  * [15] S. L. Campbell and W. Marszalek.  The index of an infinite dimensional implicit system.  Math. and Comp. Model. Dynam. Systems, 5(1):18–42, 1999.
  * [16] S.L. Campbell and W. Marszalek.  DAEs arising from traveling wave solutions of pdes.  J. Comput. Appl. Math., 82(1-2):41–58, 1997.
  * [17] F. L. Cardoso-Ribeiro, D. Matignon, and V. Pommier-Budinger.  A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system.  J. Fluids and Structures, 69:402–427, 2017.
  * [18] P. G. Ciarlet.  The finite element method for elliptic problems.  SIAM, Philadelphia, PA, 2002.
  * [19] H. Egger and T. Kugler.  Damped wave systems on networks: Exponential stability and uniform approximations.  Numer. Math., 138(4):839–867, 2018.
  * [20] E. Emmrich and V. Mehrmann.  Operator differential-algebraic equations arising in fluid dynamics.  Comput. Methods Appl. Math., 13(4):443–470, 2013.
  * [21] M. Erbay, B. Jacob, K. Morris, T. Reis, and C. Tischendorf.  Index concepts for linear differential-algebraic equations in finite and infinite dimensions.  arXiv preprint arXiv:2401.01771, 2024.
  * [22] L. C. Evans.  Partial differential equations.  American Mathematical Society, 2022.
  * [23] F. R. Gantmacher.  Theory of Matrices.  Chelsea, New York, 1959.
  * [24] H. Gernandt, F. E. Haller, and E. Reis.  A linear relation approach to port-Hamiltonian differential-algebraic equations.  SIAM J. Matrix Anal. Appl., 42(2):1011–1044, 2021.
  * [25] H. Gernandt and C. Trunk.  The spectrum and the weyr characteristics of operator pencils and linear relations.  Syphax Journal of Mathematics: Nonlinear Analysis, Operator and Systems, 1:73–89, 2021.
  * [26] B. Jacob and K. Morris.  On solvability of dissipative partial differential-algebraic equations.  arXiv preprint arXiv:2204.10599, 2022.
  * [27] B. Jacob and H. Zwart.  Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces.  Operator Theory: Advances and Applications, 223. Birkhäuser/Springer Basel AG, Basel CH, 2012.
  * [28] H. Khlif, C. Trunk, and M. Wilson.  On the essential spectrum of operator pencils.  Technical Report 23/11, TU Ilmenau, Inst. f. Mathematik, Ilmenau, 2023\.
  * [29] P. Kotyczka.  Numerical Methods for distributed parameter Port-Hamiltonian Systems. Structure-Preserving Approaches for Simulation and Control.  Habilitation thesis, TU Munich, 2019.
  * [30] Vadym Koval and Patryk Pagacz.  On singular pencils with commuting coefficients.  arXiv preprint arXiv:2402.00673, 2024.
  * [31] C. S. Kubrusly.  Spectral theory of operators on Hilbert spaces.  Springer Science & Business Media, 2012.
  * [32] P. Kunkel and V. Mehrmann.  Differential-Algebraic Equations. Analysis and Numerical Solution.  EMS Publishing House, Zürich, Switzerland, 2006.
  * [33] M. Kurula, H. Zwart, A. van der Schaft, and J. Behrndt.  Dirac structures and their composition on Hilbert spaces.  J. Math. Anal. Appl., 372(2):402–422, 2010.
  * [34] F. Laurén and J. Nordström.  Spectral properties of the incompressible Navier-Stokes equations.  J. Comp. Physics, 429:110019, 2021.
  * [35] Y. Le Gorrec, H. Zwart, and B. Maschke.  Dirac structures and boundary control systems associated with skew-symmetric differential operators.  SIAM J. Cont. Optim., 44:1864–1892, 2005.
  * [36] W. Lucht, K. Strehmel, and C. Eichler-Liebenow.  Indexes and special discretization methods for linear partial differential algebraic equations.  BIT, 39(3):484–512, 1999.
  * [37] A. Macchelli and B. M. Maschke.  Chapter 4: Infinite-dimensional Port-Hamiltonian Systems.  In V. Duindam, A. Macchelli, S. Stramigioli, and H. Bruyninckx, editors, Modeling and Control of Complex Physical Systems - The Port-Hamiltonian Approach, pages 211–272. Springer, 2009.
  * [38] A. Macchelli, C. Melchiorri, and L. Bassi.  Port-based modelling and control of the Mindlin plate.  In Proceedings of the 44th IEEE Conference on Decision and Control, pages 5989–5994, 2005.
  * [39] A. Macchelli, A. van der Schaft, and C. Melchiorri.  Port Hamiltonian formulation of infinite dimensional systems I. Modeling.  In 2004 43rd IEEE Conference on Decision and Control (CDC), volume 4, pages 3762–3767, 2004.
  * [40] A. Macchelli, A. van der Schaft, and C. Melchiorri.  Port Hamiltonian formulation of infinite dimensional systems II. Boundary control by interconnection.  In 2004 43rd IEEE Conference on Decision and Control (CDC), volume 4, pages 3768–3773, 2004.
  * [41] D. Matignon and T. Hélie.  A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems.  European J. Control, 19(6):486–494, 2013.
  * [42] C. Mehl, V. Mehrmann, and M. Wojtylak.  Linear algebra properties of dissipative Hamiltonian descriptor systems.  SIAM J. Matrix Anal. Appl., 39(3):1489–1519, 2018.
  * [43] C. Mehl, V. Mehrmann, and M. Wojtylak.  Distance problems for dissipative Hamiltonian systems and related matrix polynomials.  Linear Algebra Appl., pages 335–366, 2021.
  * [44] V. Mehrmann and B. Unger.  Control of port-Hamiltonian differential-algebraic systems and applications.  Acta Numerica, pages 395–515, 2023.
  * [45] V. Mehrmann and H. Zwart.  Abstract dissipative Hamiltonian differential-algebraic equations are everywhere.  ArXiv preprint arXiv:2311.03091, 2023.
  * [46] R. Morandin.  Modeling and numerical treatment of port-Hamiltonian descriptor systems.  Dissertation, Technische Universität Berlin, Inst. f. Mathematik MA4-5, D-10623 Berlin, 2024.
  * [47] H. C. Öttinger and M. Grmela.  Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism.  Physical Review E, 56(6):6633 – 6655, 1997.
  * [48] M. Puche, T. Reis, and F. L. Schwenninger.  Constant-coefficient differential-algebraic operators and the Kronecker form.  Linear Algebra Appl., 552:29–41, 2018.
  * [49] M. H. Ramirez Estay.  Modeling and control of irreversible thermodynamic processes and systems described by partial differential equations. A port-Hamiltonian approach.  Habilitation thesis, Université Bourgogne Franche-Comté, 2019\.
  * [50] T. Reis.  Circuit synthesis of passive descriptor systems – a modified nodal approach.  Int. J. Circuit Theory Appl., 38(1):44–68, 2010.
  * [51] T. Reis and M. Schaller.  Port-Hamiltonian formulation of oseen flows.  arXiv preprint arXiv:2305.09618, 2023.
  * [52] R. E. Showalter.  Diffusion in poro-elastic media.  J. Math. Anal. Appl., 251(1):310–340, 2000.
  * [53] H. Sohr.  The Navier-Stokes Equations: An Elementary Functional Analytic Approach.  Birkhäuser, Basel – Boston – Berlin, 2001.
  * [54] R. Temam.  Navier-Stokes Equations. Theory and Numerical Analysis.  North Holland, Amsterdam, The Netherlands, 1977.
  * [55] S. Trostorff.  Semigroups associated with differential-algebraic equations.  In Conference on Semigroups of Operators: Theory and Applications, pages 79–94. Springer, 2018.
  * [56] A. J. van der Schaft and B. M. Maschke.  Hamiltonian formulation of distributed parameter systems with boundary energy flow.  J. Geometry and Physics, 42(1–2):166–174, 2002.
  * [57] J. A. Villegas.  A port-Hamiltonian approach to distributed parameter systems.  PhD thesis, University of Twente, 2007.
  * [58] J. Weidmann.  Linear Operators in Hilbert Spaces, volume 68.  Springer Science & Business Media, 2012.