# On the Spectral Theory of Linear Differential-Algebraic Equations with
Periodic Coefficients

Bader Alshammari and Aaron Welters<EMAIL_ADDRESS>Department of
Mathematical Sciences
Florida Institute of Technology
Melbourne, FL 32937, USA

###### Abstract.

In this paper, we consider the spectral theory of linear differential-
algebraic equations (DAEs) for periodic DAEs in canonical form, i.e.,

$J\frac{df}{dt}+Hf=\lambda Wf,$

where $J$ is a constant skew-Hermitian $n\times n$ matrix that is not
invertible, both $H=H(t)$ and $W=W(t)$ are $d$-periodic Hermitian $n\times
n$-matrices with Lebesgue measurable functions as entries, and $W(t)$ is
positive semidefinite and invertible for a.e. $t\in\mathbb{R}$ (i.e., Lebesgue
almost everywhere). Under some additional hypotheses on $H$ and $W$, called
the local index-1 hypotheses, we study the maximal and the minimal operators
$L$ and $L_{0}^{\prime}$, respectively, associated with the differential-
algebraic operator $\mathcal{L}=W^{-1}(J\frac{d}{dt}+H)$, both treated as an
unbounded operators in a Hilbert space $L^{2}(\mathbb{R};W)$ of weighted
square-integrable vector-valued functions. We prove the following: (i) the
minimal operator $L_{0}^{\prime}$ is a densely defined and closable operator;
(ii) the maximal operator $L$ is the closure of $L_{0}^{\prime}$; (iii) $L$ is
a self-adjoint operator on $L^{2}(\mathbb{R};W)$ with no eigenvalues of finite
multiplicity, but may have eigenvalues of infinite multiplicity. As an
important application, we show that for 1D photonic crystals with passive
lossless media, Maxwell’s equations for the electromagnetic fields become,
under separation of variables, periodic DAEs in canonical form satisfying our
hypotheses so that our spectral theory applies to them (a primary motivation
for this paper).∗

###### Key words and phrases:

linear differential-algebraic equations with periodic coefficients, canonical
DAEs, differential-algebraic operators, functional analysis, unbounded
operators, spectral theory, self-adjoint, eigenvalue problems,
electromagnetism, photonic crystals, layered (stratified) media

###### 2020 Mathematics Subject Classification:

34A09, 34L05, 47B25, 47A75, 78M22, 46N20, 47N20, 47B38, 47B93, 47N50

*This work is based in part on the Ph.D. dissertation of the first author.

##  1\. Introduction

The spectral theory of linear differential-algebraic equations (DAEs) with
periodic coefficients is of significant interest from both a theoretical point
of view and in applications (see, for instance, [24, 11, 25, 13] and [45, Sec.
6.6.7]) and this is especially true for electromagnetic problems involving 1D
photonic crystals (see, for instance, [14, 56, 49, 50, 51]). For linear
ordinary differential equations (ODEs), the spectral theory is well-developed
(see, for instance, [12, 55]). But this is not true for DAEs when the interval
under consideration is unbounded or when the spectral problem is posed on a
Hilbert space of weighted $L^{2}$ functions. Yet, such DAEs arise naturally
when considering electromagnetic phenomena in periodic layered media (1D
photonic crystals) involving passive lossless media (especially when the
materials as anisotropic, biisotropic, or bianisotropic instead of isotropic).
In this case, solving for the time-harmonic electromagnetic fields using
separation of variables, Maxwell’s equations are reduced to exactly such a
spectral problem on the whole real line for DAEs in canonical form.

Motivated by such applications, we will consider the spectral theory for
periodic DAEs in the following implicit canonical (cf. [23]) or Hamiltonian
form:

(1.1) $J\frac{df}{dt}+Hf=\lambda Wf,$

where $J$ is a constant skew-Hermitian $n\times n$ matrix that is not
invertible (i.e., $\det J=0$), both $H=H(t)$ and $W=W(t)$ are $d$-periodic
Hermitian $n\times n$ matrices with Lebesgue measurable functions as entries,
and $W(t)$ is positive semidefinite and invertible for a.e. $t\in\mathbb{R}$
(i.e., Lebesgue almost everywhere).

In this paper we study the maximal and the minimal operators $L$ and
$L_{0}^{\prime}$ (see Def. 4.5), respectively, associated with the
differential-algebraic (DA) operator $\mathcal{L}=W^{-1}(J\frac{d}{dt}+H)$
(see Def. 4.1), both treated as an unbounded operators in a Hilbert space
$L^{2}(\mathbb{R};W)$ [as defined by (4.14) with inner product (4.15); see
also Remark 4.4] of weighted square-integrable vector-valued functions. Our
main result is Theorem 5.9 [that requires the local index-1 hypotheses (see
Def. 4.14) are true for $H-z_{0}W,W$ with respect to $J$ on the interval
$\mathbb{R}$ for some $z_{0}\in\mathbb{C}\setminus\mathbb{R}$] which can be
summarized as follows: (i) the minimal operator $L_{0}^{\prime}$ is a densely
defined and closable operator; (ii) the maximal operator $L$ is the closure of
$L_{0}^{\prime}$ (i.e, $L=\overline{L_{0}^{\prime}}$); (iii) $L$ is a self-
adjoint operator on $L^{2}(\mathbb{R};W)$ (that is, $L^{*}=L$) with no
eigenvalues of finite multiplicity, but may have eigenvalues of infinite
multiplicity [or, equivalently, $\dim\ker(L-\lambda I)\in\\{0,\infty\\}$ for
each $\lambda\in\mathbb{C}$]. In addition, we give two examples of such an $L$
that does have an eigenvalue of infinite multiplicity (see Examples 5.10 and
5.15).

An important result of this paper is Theorem 5.13 which gives simple
hypotheses on the coefficients $H,W$ in terms of $J$ that are sufficient for
Theorem 5.9 to be true [i.e., for (i)-(iii) to be true] and they can be stated
as follows: Let $v_{1},\ldots,v_{n_{1}}$ and $v_{n_{1}+1},\ldots,v_{n}$ be an
orthonormal basis of $n\times 1$ column vectors for the range of $J$ and
kernel (nullspace) of $J$, respectively, and define the $n\times n$ unitary
matrix $V$ to be the column matrix
$V=[v_{1}|\cdots|v_{n_{1}}|v_{n_{1}+1}|\cdots|v_{n}]$. We next define
$n_{2}=n-n_{1}$ and the $n_{i}\times n_{j}$ matrix-valued functions $H_{ij}$
and $W_{ij}$ (for $i,j=1,2$) by forming a $2\times 2$ block matrix
partitioning of $V^{-1}HV$ and $V^{-1}WV$ as

(1.2) $\displaystyle V^{-1}HV=\begin{bmatrix}H_{11}&H_{12}\\\
H_{21}&H_{22}\end{bmatrix},\;V^{-1}WV=\begin{bmatrix}W_{11}&W_{12}\\\
W_{21}&W_{22}\end{bmatrix}.$

Then Theorem 5.13 tells us that if all the entries of $H_{11},W_{11},$ and
$H_{12}W_{22}^{-1}H_{21}$ are (i.e., integrable on the interval $[0,d]$) then
Theorem 5.9 is true.

Let us compare all this to the spectral theory for periodic ODEs in the form
(1.1) where $J$ is invertible ($\det J\not=0$), also known as canonical or
Hamiltonian (system of) ODEs (see, for instance, [4, 58, 12, 55, 10, 47, 3,
46, 43, 5]). In [12], V. I. Derguzov proves, under the hypotheses $H(t)$ and
$W(t)$ are also smooth symmetric matrix-valued functions, that the maximal
operator $L$ is self-adjoint on $L^{2}(\mathbb{R};W)$ by giving an explicit
spectral representation of $L$ expanded in terms of the Bloch solutions of the
periodic ODEs (1.1) which proves $L$ has purely absolutely continuous (AC)
spectrum and is unitarily equivalent to multiplication by $\lambda$ in a
special Hilbert space of square-integrable functions. Furthermore, he proves
that the spectral multiplicity at each $\lambda$ is equal to the number of
linearly independent bounded solutions (corresponding to the dimension of the
space of those Bloch solutions) of (1.1). In [55], J. Weidmannn has shown
using different methods that you can weaken the hypotheses that Derguzov had
to $H$ and $W$ are Hermitian with entries in $L^{1}([0,d])$ and still prove
the maximal operator $L$ is self-adjoint on $L^{2}(\mathbb{R};W)$ [55, Theorem
12.3] with purely AC spectrum and $L=\overline{L_{0}^{\prime}}$ is the closure
of the minimal operator $L_{0}^{\prime}$ [55, Theorem 12.4]. In particular,
the self-adjoint operator $L$ has no eigenvalues. It should be pointed out
that we are not aware of any weaker hypotheses that allow such a broad class
of coefficients $H,W$ which yield these results and in this sense, [55] is the
“state-of-the-art” for the spectral theory of canonical ODEs with periodic
coefficients.

Our goal here is to develop such a spectral theory for DAEs in the form (1.1)
that is comparable to those results above for ODEs, under the weakest
hypotheses on $H,W$ we can get. Based on techniques from [55] adapted to DAEs
together with the spectral theory for unbounded operators [41, 42] and the
theory of Schur complements [64], we are able to prove the results (i)-(iii)
in Theorem 5.9 which represent significant progress toward this goal.
Furthermore, as the local index-$1$ hypotheses in that theorem may not be easy
to verify in applications, we also provide Proposition 5.12 and Theorem 5.13
to help simplify those hypotheses. Finally, given the difficulties of proving
our main results (i)–(iii), we do not try to prove the self-adjoint operator
$L$ has no singular continuous spectrum (as done for ODEs in [12] and [55])
nor do we try to develop an analog spectral representation of $L$ (as was done
in [12] for ODEs) in terms of an expansion in Bloch solutions. Nevertheless,
for a complete spectral theory on $L$, it would be desirable to do so for the
applications mentioned above, but this is left for future work.

As a final remark, the literature on DAEs using functional analysis and
operator theory that is closest in spirit to our approach, especially Section
4, is [16, 65, 34, 17]. They are mainly interested in closedness and normal
solvability of the maximal operator $L$ on bounded or compact intervals $I$
for Hilbert spaces of $L^{2}$ functions and its adjoint relationship to the
minimal operator $L_{0}^{\prime}$. However, they do not treat the self-adjoint
spectral theory of DAEs nor do they treat unbounded intervals nor weighted
Hilbert spaces of $L^{2}$ functions as we need to in this paper. In addition,
if we were to use the approach that they do then the weakest hypotheses on
$H,W$ that we could use would require its entries to be in $L^{\infty}$ (i.e.,
essentially bounded functions) among other things. Nevertheless, these works
are complementary to our paper.

The rest of this paper is organized as follows. In Section 2, we motivate our
studies on the spectral theory of periodic DAEs by considering
electromagnetism and Maxwell’s equations associated with 1D photonic crystals
and passive lossless media. In Section 3, we briefly introduce some basic
notations, conventions, and definitions for matrices, operators, and functions
spaces needed. We also include some preliminary results from the elementary
theory of linear ODEs and spectral theory for unbounded operators. In Section
4, we study the differential operators $L_{0}^{\prime},L,\mathcal{L}$
associated with the DAEs on an arbitrary nonempty interval
$I\subseteq\mathbb{R}$ (both for bounded as well as unbounded intervals). In
Section 5, we study the periodic case of these operators (with $I=\mathbb{R}$)
and prove our main results (i)–(iii) above. Finally, in Subsec. 5.1, we show
how to simplify the hypotheses needed to prove those results.

##  2\. One-dimensional photonic crystals

In this section, we motivate our studies on periodic linear DAEs by showing
how they arise in electromagnetism (EM) involving 1D photonic crystals. In
particular, starting from Maxwell’s equations, we consider time-harmonic
electromagnetic fields in a periodic layered media (i.e., a 1D photonic
crystal) with passive lossless materials, and show how to proceed to reduce
the problem from one involving partial differential equations (PDEs) to
periodic linear DAEs in the canonical form (1.1).

To set up the problem, we recall that Maxwell’s equations describe the
electric and magnetic fields arising from distributions of electric charges
and currents, and how those fields change in time (for more details, see [18,
26, 22, 20]). Mathematically, the (macroscopic) Maxwell’s equations (in
Gaussian units) are the PDEs:

(2.1) $\displaystyle\nabla\times E=-\frac{1}{c}\frac{\partial B}{\partial
t},\qquad\nabla\times H=\frac{4\pi}{c}J+\frac{1}{c}\frac{\partial D}{\partial
t},$ (2.2) $\displaystyle\nabla\cdot D=4\pi\rho,\qquad\nabla\cdot B=0,$

where $E$ and $H$ are the (macroscopic) electric and magnetic fields, $D$ and
$B$ are the electric displacement and magnetic induction fields, $\rho$ and
$J$ are the free charge and current densities (with $\rho=0,J=0$ in there are
no sources), and $c$ is the speed of light (in a vacuum).

For passive lossless (linear) homogeneous media, the constitutive relations
are of the form

(2.3) $\displaystyle D=\varepsilon E+\xi H,$ (2.4) $\displaystyle B=\zeta
E+\mu H,$

where $\varepsilon,\mu,\xi,\mu\in M_{3}(\mathbb{C})$ (dielectric permittivity,
magnetic permeability, and magnetoelectric coupling tensors), i.e., are
$3\times 3$ matrices with entries in $\mathbb{C}$, such that the $2\times 2$
block matrix $M=[M_{ij}]_{i,j=1,2}\in M_{6}(\mathbb{C})$ defined by

(2.5) $\displaystyle M=\begin{bmatrix}\varepsilon&\xi\\\
\zeta&\mu\end{bmatrix},$

satisfies the following self-adjoint and positivity constraints (see, for
instance, [26, Sec. 80], [31, 57]):

(2.6) $\displaystyle M^{*}=M>0,$

(i.e., $M$ is a Hermitian matrix that is positive semidefinite and
invertible). In particular, we have

(2.7)
$\displaystyle\varepsilon^{*}=\varepsilon,\;\mu^{*}=\mu,\;\zeta=\xi^{*}.$

The media can now be classified as follows (cf. [22, 30, 52, 53]). In the case
$\xi=0$, the media is called isotropic if $\varepsilon$ and $\mu$ are scalar
matrices (i.e., scalar multiples of $I_{3}$, the $3\times 3$ identity matrix)
and if not then it is anisotropic. In the case $\xi\not=0$, the media is
called biisotropic if $\varepsilon,\mu,\xi$ are scalar matrices and if not
then it is bianisotropic.

We consider now a one-dimensional (1D) photonic crystal made of plane parallel
layers of these materials. More precisely, a $d$-periodic layered (or
stratified) medium whose layers are normal to $z$-axis, homogeneous in the
$xy$-plane, and consisting only of passive linear lossless layers for a.e.
$z\in\mathbb{R}$, i.e., the constitutive relations are of the form (2.3, 2.4),
but now the matrix $M$ in (2.5) is a $6\times 6$ matrix-valued function
$M=M(z)\in M_{6}(\mathcal{M}(\mathbb{R}))$ satisfying

(2.8) $\displaystyle M(z+d)=M(z),\;\;M(z)^{*}=M(z)>0\text{ for a.e.
}z\in\mathbb{R}.$

Then Maxwell’s equations without sources (i.e., $\rho=0,J=0$) for the
electromagnetic fields $E,H$ in this medium admit time-harmonic solutions of
the form

(2.9) $\displaystyle E=E(z)e^{i(k_{1}x+k_{2}y-\omega
t)},\;H=H(z)e^{i(k_{1}x+k_{2}y-\omega t)},$

where $\omega$ is the frequency, $i=\sqrt{-1}$, and
$(k_{1},k_{2})\in\mathbb{R}^{2}$ is the wavevector parallel to the layers.
Using separation of variables, Maxwell’s equations (without sources) for these
time-harmonic solutions are reduced to the periodic Maxwell’s DAEs:

(2.10) $J\frac{d}{dz}f+Hf=\lambda Wf,$ (2.10a)
$\displaystyle\lambda=\frac{\omega}{c},$ (2.10b) $\displaystyle
f(z)=\begin{bmatrix}E(z)\\\ H(z)\end{bmatrix},$ (2.10c) $\displaystyle
J=i^{-1}\begin{bmatrix}0&-e_{3}\times\\\ e_{3}\times&0\\\
\end{bmatrix},\,e_{3}\times=\begin{bmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&0\\\
\end{bmatrix},$ (2.10d) $\displaystyle H=\begin{bmatrix}0&k_{\bot}\times\\\
-k_{\bot}\times&0\\\
\end{bmatrix},\,k_{\bot}\times=\begin{bmatrix}0&0&k_{2}\\\ 0&0&-k_{1}\\\
-k_{2}&k_{1}&0\\\ \end{bmatrix},$ (2.10e) $\displaystyle
W=M=\begin{bmatrix}\varepsilon&\xi\\\ \zeta&\mu\end{bmatrix}.$

In particular, these are periodic DAEs in canonical form (1.1) (with the
independent variable $t$ replaced by $z$) such that $J$ is a constant skew-
Hermitian $6\times 6$ matrix that is not invertible (i.e., $\det J=0$), both
$H$ (a constant function of $z$) and $W=W(z)$ are $d$-periodic Hermitian
$6\times 6$ matrices with Lebesgue measurable functions as entries, and $W(z)$
is positive semidefinite and invertible for a.e. $z\in\mathbb{R}$.

Two other types of applied electromagnetic problems for 1D photonic crystals
that are worth briefly mentioning which have the canonical form (2.10), where
$f,J$ remains the same, but the $\lambda,H,W$ (assuming $\omega>0$) change as
follows.

  * •

Disorder added to the periodic unit cell: e.g.,
$\varepsilon(\lambda):=\varepsilon_{0}+\lambda\varepsilon_{1},$
$\mu(\lambda):=\mu_{0}+\lambda\mu_{1}$ ($\lambda$ scales amount of disorder),

$\displaystyle H=\left(\begin{array}[]{cc}0&k_{\bot}\times\\\
-k_{\bot}\times&0\\\
\end{array}\right)-\frac{\omega}{c}\left(\begin{array}[]{cc}\varepsilon_{0}&0\\\
0&\mu_{0}\\\
\end{array}\right),\;\;W=\frac{\omega}{c}\left(\begin{array}[]{cc}\varepsilon_{1}&0\\\
0&\mu_{1}\\\ \end{array}\right).$

  * •

Lossy photonic crystals: e.g.,
$\varepsilon(\gamma):=\operatorname{Re}\varepsilon+i\gamma\operatorname{Im}\varepsilon$,
$\mu(\gamma):=\operatorname{Re}\mu+i\gamma\operatorname{Im}\mu$, $|\gamma|\ll
1$ (low-loss), $|\gamma|\gg 1$ (high-loss), $\lambda:=i\gamma$ ($\lambda$
scales amount of loss),

$\displaystyle H=\left(\begin{array}[]{cc}0&k_{\bot}\times\\\
-k_{\bot}\times&0\\\
\end{array}\right)-\frac{\omega}{c}\left(\begin{array}[]{cc}\operatorname{Re}\varepsilon&0\\\
0&\operatorname{Re}\mu\\\
\end{array}\right),\;\;W=\frac{\omega}{c}\left(\begin{array}[]{cc}\operatorname{Im}\varepsilon&0\\\
0&\operatorname{Im}\mu\\\ \end{array}\right).$

In summary, the electrodynamics of 1D photonic crystals in passive linear
lossless media provides a model that motivates this paper on the spectral
theory of periodic DAEs in the canonical form (1.1). For as we have shown in
this model, Maxwell’s equations become the periodic DAEs (2.10) in this form
and so the spectral theory developed in this paper applies to these DAEs (a
primary motivation for this paper).

Now before we move on, let us use this example to demonstrate here how our
approach would begin when we want to solve DAEs by reducing them to ODEs
[which is part of the reason why we need the (local) index-$1$ hypotheses]. As
$\det J=0$, let us introduce the unitary matrix $V\in M_{6}(\mathbb{C})$,
i.e.,

(2.11) $\displaystyle V\in M_{6}(\mathbb{C}),\;\;V^{*}=V^{-1},$

defined by

(2.12) $\displaystyle V=\begin{bmatrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&0&0&1&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&0&1\\\ \end{bmatrix}.$

Then $V^{-1}JV=[J_{ij}]_{i,j=1,2}$ has the $2\times 2$ block partitioned
matrix form

(2.13) $\displaystyle
V^{-1}JV=[J_{ij}]_{i,j=1,2}=\begin{bmatrix}J_{11}&J_{12}\\\ J_{21}&J_{22}\\\
\end{bmatrix}=\begin{bmatrix}J_{11}&0\\\ 0&0\\\
\end{bmatrix},\quad\det(J_{11})\neq 0,$

and, more precisely,

(2.14) $\displaystyle J_{ij}\in M_{n_{i}\times
n_{j}}(\mathbb{C}),i,j=1,2;\;\;J_{ij}=0,\;\;(i,j)\not=(1,1),\;\;\det(J_{11})\neq
0,$

where we define $n_{1},n_{2}\in\mathbb{N}$ by

(2.15) $\displaystyle n_{1}:=\operatorname{rank}J=4,\;\;n_{2}:=\dim\ker
J=\operatorname{nullity}(J)=6-n_{1}=2.$

In particular, $J_{11}\in M_{4}(\mathbb{C})$ is given by

(2.16) $\displaystyle J_{11}=i^{-1}\begin{bmatrix}0&0&0&1\\\ 0&0&-1&0\\\
0&-1&0&0\\\ 1&0&0&0\\\ \end{bmatrix},\;i=\sqrt{-1}.$

The relevance of this matrix $V$ is as follows. Consider the components of the
electromagnetic fields

(2.17) $\displaystyle
E=\begin{bmatrix}E_{1}&E_{2}&E_{3}\end{bmatrix}^{T},\;H=\begin{bmatrix}H_{1}&H_{2}&H_{3}\end{bmatrix}^{T}.$

Then in terms of the tangential components (i.e., those with index $i=1,2$)
and the normal components (i.e., those index by $i=3$) we have

(2.18) $\displaystyle
V^{-1}\begin{bmatrix}E&H\end{bmatrix}^{T}=\begin{bmatrix}E_{1}&E_{2}&H_{1}&H_{2}&E_{3}&H_{3}\end{bmatrix}^{T}.$

Hence, for the time-harmonic fields $E,H$ of the form (2.9) and in terms of
the corresponding $6\times 1$ column vector-valued function
$f(z)=\begin{bmatrix}E(z)&H(z)\end{bmatrix}^{T}$ (written as in a $2\times 1$
block vector form), the vector function $V^{-1}f$ has the following $2\times
1$ block vector form:

(2.19) $\displaystyle V^{-1}f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},\;f(z)=\begin{bmatrix}E(z)\\\
H(z)\end{bmatrix},\;f_{1}(z)=\begin{bmatrix}E_{1}(z)\\\ E_{2}(z)\\\
H_{1}(z)\\\ H_{2}(z)\end{bmatrix},\;f_{2}(z)=\begin{bmatrix}E_{3}(z)\\\
H_{3}(z)\end{bmatrix}.$

The importance of introducing $V$ and $J_{11}$ is then the following: The
tangential components
$f_{1}(z)=\begin{bmatrix}E_{1}(z)&E_{2}(z)&H_{1}(z)&H_{2}(z)\end{bmatrix}^{T}$
give rise to the differential part of the periodic Maxwell’s DAEs (2.10) (see
also Example 4.3), that is, they are the solutions of a linear system of
canonical ODEs with periodic coefficients in which $J_{11}$ is now the matrix
coefficient for the derivative term, whereas the normal components
$f_{2}(z)=\begin{bmatrix}E_{3}(z)&H_{3}(z)\end{bmatrix}^{T}$ give rise to the
algebraic part of (2.10), that is, they are the output of left multiplication
by a matrix function of $z$ acting on $f_{1}(z)$. This statement can be made
rigorous using Lemma 4.2 and Proposition 4.17 [by taking the interval
$I=\mathbb{R}$ and the coefficients $J,H,W$ to be (2.10c), (2.10d), (2.10e)],
two key results in this paper on solvability of linear DAEs.

We conclude this section with a comment and then two remarks. One thing to
keep in mind here is that the example above demonstrates the complexities that
arise when considering linear DAEs versus linear ODEs, but also shows that
there is a need to develop the operator and spectral theory for them. But we
do have simpler examples in this paper for which we apply our theory, see
Examples 5.10 and 5.15.

######  Remark 2.1.

The methods we use in this paper to reduced linear DAEs to linear ODEs is
implicit in a standard and well known approach in studying the electrodynamics
of layered media [7, 1, 2, 14, 38, 40, 32, 33, 39] (as opposed to other
standard methods such as those in [62, 59, 60, 61]) and has been used
effectively by the second author of this paper, for instance, in [56, 50, 51,
8]. But in this context, our approach that we’ve developed in the sections
below, using DA operators and spectral theory, is completely new and deserves
to be further studied from this perspective. Our paper represents a first step
in this direction of research.

######  Remark 2.2.

We are motivated to study the spectral theory of linear DAEs because we are
interested specifically in electromagnetic applications involving 1D photonic
crystals. More broadly, it is known that linear DAEs arise in computational
electromagnetics [9] and in circuit theory [44], and as such it would be
interesting and worth investigating how our methods and results would be
useful in these contexts too.

##  3\. Preliminaries

This paper lies at the intersection of the several different areas, namely,
ODEs and DAEs, spectral theory for unbounded operators, and electromagnetics
(as discussed in the previous section). As such, we expect that the reader’s
level of expertise on these topics may widely vary. Thus, we try to give here
some necessary background to make our paper more approachable and self-
contained. First, we will introduce the standard function spaces that are
needed for describing existence and uniqueness theorems on initial-value
problems (IVPs) for linear ODEs/DAEs. After this we recall, for unbounded
operators on Hilbert spaces, the following classes of operators: densely
defined, closable, closed, symmetric, and self-adjoint.

###  3.1. Function spaces

Let $\mathbb{N},\mathbb{R},\mathbb{C}$ denote the natural, real, and complex
numbers, respectively. For $-\infty\leq a<b\leq\infty,$ an open, closed, or
half-open interval $I\subseteq\mathbb{R}$ is denoted by $(a,b),$ $[a,b],$ and
$[a,b),(a,b]$, respectively; if $a=-\infty$ or $b=\infty$, then the interval
is unbounded, otherwise it is bounded (e.g., compact intervals are precisely
those intervals that are closed and bounded) and, as usual,
$[a,\infty]=[a,\infty),(a,\infty]=(a,\infty),[-\infty,b]=(-\infty,b],[-\infty,b)=(-\infty,b)$.

For any $z\in\mathbb{C}$, we denote its complex conjugate and (Hilbert space)
norm by $\overline{z}$ and $|z|=(\overline{z}z)^{1/2},$ respectively.

For any interval $I$, the complex vector space of Lebesgue measurable
functions (with equality in the sense of equal a.e. on $I$) is denoted by
$\mathcal{M}(I)$ [if $I=(a,b)$ then instead of writing $(I)=((a,b))$ we drop
the extra parentheses, e.g., $\mathcal{M}(I)=\mathcal{M}(a,b)$]; for each
$p\in[1,\infty)$, the subspace $L^{p}(I)$ of $\mathcal{M}(I)$ is defined by

(3.1) $\displaystyle
L^{p}(I)=\left\\{f\in\mathcal{M}(I):\int_{I}|f(t)|^{p}dt<\infty\right\\},$

which is a Banach space with norm

(3.2) $\displaystyle||f||_{p}=\left(\int_{I}|f(t)|^{p}dt\right)^{1/p},\;\;f\in
L^{p}(I)$

and, in the case $p=2$, is a Hilbert space with inner product

(3.3) $\displaystyle\langle
f,g\rangle_{2}=\int_{I}\overline{f(t)}g(t)dt,\;\;f,g\in L^{2}(I);$

the subspace $L^{\infty}(I)$ of $\mathcal{M}(I)$ is defined in terms of the
essential supremum by

(3.4) $\displaystyle
L^{\infty}(I)=\left\\{f\in\mathcal{M}(I):\operatorname*{ess\,sup}_{t\in
I}|f(t)|<\infty\right\\},$

which is a Banach space with norm

(3.5) $\displaystyle||f||_{\infty}=\operatorname*{ess\,sup}_{t\in
I}|f(t)|,\;\;f\in L^{\infty}(I).$

For any compact interval $I=[a,b]$, we denote the Banach space of all complex-
valued absolutely continuous functions on the interval $I$ by $AC(I)$ with
norm

(3.6)
$\displaystyle||f||_{AC(I)}=|f(a)|+\int_{I}\left|\frac{df}{dt}(\tau)\right|d\tau,\;\;f\in
AC(I).$

For any interval $I$, the subspace $L_{loc}^{p}(I)$ of $\mathcal{M}(I)$ and
the complex vector space $AC_{loc}(I)$ are defined by

(3.7) $\displaystyle L_{loc}^{p}(I)=\\{f\in\mathcal{M}(I):f\in
L^{p}([a,b]),\text{ for every compact interval }[a,b]\subseteq I\\},$ (3.8)
$\displaystyle AC_{loc}(I)=\\{f:I\rightarrow\mathbb{C}\;|\;f\in
AC([a,b]),\text{ for every compact interval }[a,b]\subseteq I\\},$

respectively. In particular,

(3.9) $\displaystyle L_{loc}^{p}(I)=L^{p}(I),AC_{loc}(I)=AC(I),\text{ if
}I\text{ is a compact interval.}$

The subspace $W^{1,p}(I)$ of $L^{p}(I)$ is defined by

(3.10) $\displaystyle W^{1,p}(I)=\left\\{f\in L^{p}(I):f\in
AC_{loc}(I),\frac{df}{dt}\in L^{p}(I)\right\\},$

which is a Banach space with norm

(3.11)
$\displaystyle||f||_{1,p}=||f||_{p}+\left\|\frac{df}{dt}\right\|_{p},\;\;f\in
W^{1,p}(I).$

The subspace $W^{1,p}_{loc}(I)$ of $L^{p}_{loc}(I)$ is defined by

(3.12) $\displaystyle W^{1,p}_{loc}(I)=\\{f\in\mathcal{M}(I):f\in
W^{1,p}([a,b]),\text{ for every compact interval }[a,b]\subseteq I\\},$

in particular,

(3.13) $\displaystyle W^{1,p}_{loc}(I)=W^{1,p}(I),\text{ if }I\text{ is a
compact interval.}$

Note that for each $f\in W_{loc}^{1,p}(I)$, there is a unique $g\in
AC_{loc}(I)$ such that $f(t)=g(t)$ for a.e. $t\in I$, and as such, we will
always use this representative of $f$ when we evaluate $f$ at a point, i.e.,
for each $t_{0}\in I$ will define $f(t_{0}):=g(t_{0})$. With this convention
we have the integral representation

(3.14) $\displaystyle
f(t)=f(t_{0})+\int_{t_{0}}^{t}\frac{df}{dt}(\tau)d\tau,\;\;t_{0},t\in I,$

and as such $f$ is a continuous function on the interval $I$. Similarly, if
$f\in W^{1,p}(I)$ then $f$ has a representation in $AC_{loc}(I)$ which has the
integral representation (3.14) and is continuous on $I$. Using this
identification of such functions with their integral representations, we can
abuse notation and make the identification

(3.15) $\displaystyle W^{1,1}(I)\subseteq W^{1,1}_{loc}(I)=AC_{loc}(I),\text{
for any interval $I\subseteq\mathbb{R}$;}$ (3.16) $\displaystyle
W^{1,1}(I)=W^{1,1}_{loc}(I)=AC_{loc}(I)=AC(I),\text{ if $I$ is a compact
interval.}$

Let $m,n\in\mathbb{N}$, $I$ an interval, $p\in[1,\infty]$, and

(3.17)
$\displaystyle\mathcal{V}\in\\{\mathbb{C},\;\mathcal{M}(I),\;L_{loc}^{p}(I),\;W^{1,p}_{loc}(I),\;AC_{loc}(I)\\}.$

We denote the set of all $n\times m$ matrices with entries in $\mathcal{V}$ by
$M_{n,m}(\mathcal{V})$, and define

(3.18)
$\displaystyle\mathcal{V}^{n}=M_{n,1}(\mathcal{V}),\;\;M_{n}(\mathcal{V})=M_{n,n}(\mathcal{V})$

and identify $\mathcal{V}$ with $\mathcal{V}^{1}.$ If $\mathcal{V}$ is a
Banach space (Hilbert space) with norm $||\cdot||_{\mathcal{V}}$ then
$M_{n,m}(\mathcal{V})$ will denote the Banach space (Hilbert space) with norm

(3.19)
$\displaystyle||[a_{ij}]||=\left(\sum_{i=1}^{n}\sum_{j=1}^{m}||a_{ij}||_{\mathcal{V}}^{2}\right)^{1/2},\;\;[a_{ij}]\in
M_{n,m}(\mathcal{V}).$

Any matrix-valued function $A:I\rightarrow M_{n,m}(\mathbb{C})$ can be written
uniquely as $A=[a_{ij}]$, where $a_{ij}:I\rightarrow\mathbb{C}$ and
$[a_{ij}](t)=[a_{ij}(t)]$ for all $t\in I$ and all
$i=1,\ldots,n,j=1,\ldots,m$; the complex conjugation
$\overline{A}:I\rightarrow M_{n,m}(\mathbb{C})$, transpose $A^{T}:I\rightarrow
M_{m,n}(\mathbb{C})$, and conjugate transpose $A^{*}:I\rightarrow
M_{m,n}(\mathbb{C})$ are defined by

(3.20)
$\displaystyle\overline{A}(t)=\overline{A(t)}=[\overline{a_{ij}(t)}],\;\;A^{T}(t)=A(t)^{T}=[a_{ji}(t)],\;\;A^{*}(t)=A(t)^{*}=[\overline{a_{ji}(t)}],\;\;\forall
t\in I.$

Similarly, the complex conjugation, transpose, and conjugate transpose of any
$[a_{ij}]\in M_{n,m}(\mathcal{V})$ satisfies

(3.21) $\displaystyle\overline{[a_{ij}]}=[\overline{a_{ij}}]\in
M_{n,m}(\mathcal{V}),\;\;[a_{ij}]^{T}=[a_{ji}],\;\;[a_{ij}]^{*}=[\overline{a_{ji}}]\in
M_{m,n}(\mathcal{V}).$

In particular, for the Hilbert spaces $\mathbb{C}^{n}$ and $(L^{2}(I))^{n}$,
their inner products $\langle\cdot,\cdot\rangle$ and
$\langle\cdot,\cdot\rangle_{2}$, respectively, are defined as

(3.22) $\displaystyle\langle x,y\rangle=x^{*}y,\;\;x,y\in\mathbb{C}^{n},$
(3.23) $\displaystyle\langle f,g\rangle_{2}=\int_{I}\langle f(t),g(t)\rangle
dt,\;\;f,g\in(L^{2}(I))^{n}.$

If $\mathcal{V}$ is a functional space in which integration
$\int_{U}(\cdot)\;dt$ over an interval $U\subseteq I$ is well-defined then we
define the integration $\int_{U}(\cdot)\;dt$ on $M_{n,m}(\mathcal{V})$ by

(3.24)
$\displaystyle\int_{U}[a_{ij}](t)dt=\left[\int_{U}a_{ij}(t)dt\right],\;\;[a_{ij}]\in
M_{n,m}(\mathcal{V}).$

Similarly, if $\mathcal{V}$ is a functional space in which differentiation
$\frac{d}{dt}$ is defined (either in classical or weak sense) then we define
differentiation $\frac{d}{dt}$ on $M_{n,m}(\mathcal{V})$ by

(3.25)
$\displaystyle\frac{d[a_{ij}]}{dt}=\left[\frac{da_{ij}}{dt}\right],\;\;\;\;[a_{ij}]\in
M_{n,m}(\mathcal{V}).$

The following well-known theorem [63] will be crucial in our paper.

######  Theorem 3.1 (ODE IVP – Existence and uniqueness of solutions).

Let $I$ be any interval and $m,n\in\mathbb{N}$. If

(3.26) $\displaystyle A\in M_{n}(L^{1}_{loc}(I)),\;\;F\in
M_{n,m}(L^{1}_{loc}(I))$

then every initial-value problem (IVP)

(3.27) $\displaystyle\frac{dX}{dt}+AX=F,$ (3.28) $\displaystyle
X(t_{0})=C,\;\;t_{0}\in I,\;\;C\in M_{n,m}(\mathbb{C})$

on $I$, has a unique solution

(3.29) $\displaystyle X\in M_{n,m}(W^{1,1}_{loc}(I)).$

Similarly, in the case in which the interval $I$ is bounded, the statement is
true if the “loc” is dropped in the hypotheses (3.26) and in the conclusion
(3.29).

###  3.2. Unbounded operators on Hilbert spaces

Here we recall from [42] some of the basic notions, definitions, and results
for unbounded operators on densely defined, closed and closable operators, and
adjoints in Hilbert spaces that will be used in this paper. All proofs of the
statements below can be found in [42] and so are omitted with the exception of
Lemma 3.10 and Theorem 3.11 (which are crucial to our paper, interesting and
useful in their own right, and deserve simple proofs).

######  Definition 3.2.

A (linear) operator $T$ on a Hilbert space $\mathcal{H}$ is a linear map from
its domain, a linear subspace of $\mathcal{H}$, into $\mathcal{H}$. This
subspace, which we denote by $D(T)$, is called the domain of the operator $T$.
Moreover, $T$ is said to be densely defined if $D(T)$ is dense in
$\mathcal{H}$ [i.e., $\overline{D(T)}=\mathcal{H}$].

######  Definition 3.3.

The graph $\Gamma(T)$ of an operator $T:D(T)\rightarrow\mathcal{H}$ is the set

$\Gamma(T)=\left\\{(\varphi,T\varphi):\varphi\in
D(T)\right\\}\subseteq\mathcal{H}\times\mathcal{H},$

and $T$ is said to be a closed operator if $\Gamma(T)$ is a closed set [i.e.,
$\overline{\Gamma(T)}=\Gamma(T)$] in the Hilbert space
$\mathcal{H}\times\mathcal{H}$ with inner product

$\langle(\varphi_{1},\psi_{1}),(\varphi_{2},\psi_{2})\rangle=\langle\varphi_{1},\psi_{1}\rangle+\langle\varphi_{2},\psi_{2}\rangle,\quad\varphi_{i},\psi_{i}\in\mathcal{H},i=1,2.$

######  Definition 3.4.

Let $T$ and $T_{1}$ be operators on a Hilbert space $\mathcal{H}$. If
$\Gamma(T)\subseteq\Gamma(T_{1})$, then $T_{1}$ is said to be extension of $T$
and we write $T\subset T_{1}$. Equivalently, $T\subset T_{1}$ if and only if
$D(T)\subseteq D(T_{1})$ and $T_{1}\varphi=T\varphi$ for all $\varphi\in
D(T).$

######  Definition 3.5.

An operator $T$ is closable if it has a closed extension. Moreover, every
closable operator $T$ has smallest closed extension called its closure, which
is denoted by $\overline{T}$.

######  Definition 3.6.

Let $T$ be a densely defined linear operator on a Hilbert space $\mathcal{H}$.
Let $D(T^{*})$ be the set of $\varphi\in\mathcal{H}$ for which there is an
$\eta\in\mathcal{H}$ with

$\langle T\psi,\varphi\rangle=\langle\psi,\eta\rangle\text{ for all }\psi\in
D(T).$

For each such $\varphi\in D(T^{*})$, we define $T^{*}\varphi=\eta$. This
linear operator $T^{*}:D(T^{*})\rightarrow\mathcal{H}$ is called the adjoint
of $T$.

It is useful to note that, unlike the case of bounded operators, the domain of
$T^{*}$ may not be dense [for instance, it is possible to have
$D(T^{*})=\\{0\\}$]. But, if $T^{*}$ is densely defined, then $T^{*}$ also has
an adjoint $(T^{*})^{*}$ which we abbreviate by $T^{**}$, i.e,
$T^{**}=(T^{*})^{*}$. Also, in terms of extensions and adjoints, if $S,T$ are
densely defined then $S\subset T$ implies $T^{*}\subset S^{*}$. The next
theorem (see [42, Theorem VIII.1]) gives a simple and useful relationship
between the notions of adjoint and closure.

######  Theorem 3.7.

Let $T$ be a densely defined operator on a Hilbert space $\mathcal{H}$. Then:

  * (a)

$T^{*}$ is closed.

  * (b)

$T$ is closable if and only if $D(T^{*})$ is dense in which case
$\overline{T}=T^{**}$.

  * (c)

If $T$ is closable, then $(\overline{T})^{*}=T^{*}$.

######  Definition 3.8.

A densely defined operator $T$ on a Hilbert space is called symmetric if
$T\subset T^{*}$, that is, if $D(T)\subseteq D(T^{*})$ and
$T\varphi=T^{*}\varphi$ for all $\varphi\in D(T)$. Equivalently, $T$ is
symmetric if and only if

(3.30) $\displaystyle\langle
T\varphi,\psi\rangle=\langle\varphi,T\psi\rangle,\text{ for all
}\varphi,\psi\in D(T).$

An operator $T$ is called self-adjoint if $T=T^{*}$, that is, if and only if
$T$ is symmetric and $D(T)=D(T^{*})$.

######  Remark 3.9.

We want to clear up some confusion that can occur for unbounded operators
between the notions of Hermitian, symmetric, and self-adjoint operators (see
[29, p. 414] for an amusing historical anecdote on this). A symmetric operator
is, by our definition (from [42]), densely defined. In [42], no distinction is
made between symmetric and Hermitian operators (see Sec. VIII.2, p. 255 in
[42]). However, in [54], a distinction is made between the two (see Sec. 4.4,
p. 72 in [54]). Specifically, in [54], a linear operator $T$ satisfying
(3.30), but not necessarily densely defined, is called Hermitian (or formally
self-adjoint, in accordance with the definition of formal adjoint in [54], see
Sec. 4.4, p. 67) and if, in addition, it is densely defined then it is called
symmetric, and if, in addition, it equals its adjoint then it is called self-
adjoint. For these definitions from [54], these classes are different, but
only for unbounded operators (i.e., examples exist of unbounded operators that
are Hermitian but not symmetric, and ones that are symmetric but not self-
adjoint).

We will need the following notation in this paper. For any linear operator
$T:V\rightarrow W$ between vector spaces $V$ and $W$ (over a common field
$F$), we denote its kernel (i.e., nullspace) and range by $\ker(T)$ and
$\operatorname{ran}(T)$, respectively. Similarly, if $A$ is a matrix with
entries in a field then we denote its kernel and range by $\ker(A)$ and
$\operatorname{ran}(A)$ [more precisely, if $T_{A}$ denotes the operator of
left-multiplication by $A$ then
$\ker(A)=\ker(T_{A}),\operatorname{ran}(A)=\operatorname{ran}(T_{A})$].

The next theorem from [48] will be useful when we discuss the adjoints of the
minimal and maximal operators associated with linear DAEs. As the proof in
[48] uses additional results not presented above and as this theorem is so
fundamental to our paper (as well as being interesting and useful more
generally), we give an alternative elementary proof of it here. To do so, we
first need the next lemma.

######  Lemma 3.10.

Let $X,Y$ be vector spaces, $D(S)$ and $D(T)$ subspaces of $X$, and
$S:D(S)\rightarrow Y$, $T:D(T)\rightarrow Y$ be linear operators satisfying
$D(S)\subseteq D(T)$ and $Sx=Tx$ for every $x\in D(S)$. Then $S=T$ if and only
if $\ker S=\ker T$ and $\operatorname{ran}S=\operatorname{ran}T$.

###### Proof.

Assume the hypotheses. If $S=T$ then obviously $\ker S=\ker T$ and
$\operatorname{ran}S=\operatorname{ran}T$. Conversely, suppose $\ker S=\ker T$
and $\operatorname{ran}S=\operatorname{ran}T$. To prove $S=T$ it suffices to
prove that $D(T)\subseteq D(S)$. Let $x\in D(T).$ Then, as
$\operatorname{ran}S=\operatorname{ran}T$, there exists $u\in D(S)$ such that
$Tx=Su$ and hence by hypotheses, $Tx=Su=Tu$. Thus, by linearity it follows
that $x-u\in\ker T$ and since $\ker T=\ker S\subseteq D(S)$ and $D(S)$ a
subspace of $X$, then $x=x-u+u\in D(S)$. This proves $D(T)\subseteq D(S)$
which completes proof. ∎

In the following theorem, we will consider the more general notion of linear
operators $A,B$ between two Hilbert spaces $H,K$ that are densely defined
closed operators which are adjoints of each other. We have already defined
these notions above in the case $H=K$, and definitions are similar in the case
$H\not=K$ so their definitions are omitted (but can be found in [54], for
instance).

######  Theorem 3.11.

If $H,K$ are Hilbert spaces with inner products
$\langle\cdot,\cdot\rangle_{H}$, $\langle\cdot,\cdot\rangle_{K}$,
respectively, $D(A)$ and $D(B)$ are subspaces of $H$ and $K$, respectively,
and $A:D(A)\rightarrow K$, $B:D(B)\rightarrow H$ are linear operators
satisfying

(3.31) $\displaystyle\langle Ax,y\rangle_{K}=\langle
x,By\rangle_{H},\;\;\text{for all }x\in D(A),y\in D(B),$ (3.32)
$\displaystyle\ker A+\operatorname{ran}B=H,$ (3.33) $\displaystyle\ker
B+\operatorname{ran}A=K,$

then $A$ and $B$ are densely defined closed operators with closed ranges and
are adjoints of each other, i.e.,

(3.34) $\displaystyle A^{*}=B,\;\;B^{*}=A.$

###### Proof.

Assume the hypotheses. First, as $A:D(A)\rightarrow K$, $B:D(B)\rightarrow H$
are linear operators then $\ker A,\operatorname{ran}B$ and $\ker
B,\operatorname{ran}A$ are subspaces of $H$ and $K$, respectively. Next, we
claim that the following orthogonality relations hold:

(3.35) $\displaystyle\ker A=(\operatorname{ran}B)^{\perp},\;\;\ker
B=(\operatorname{ran}A)^{\perp},$ (3.36) $\displaystyle(\ker
A)^{\perp}=\operatorname{ran}B,\;\;(\ker B)^{\perp}=\operatorname{ran}A.$

We now prove this claim. Let $x\in(\operatorname{ran}B)^{\perp}.$ Then, by
(3.32), $x=u+By$ for some $u\in\ker A,y\in D(B)$ and hence together with
(3.31) we have

(3.37) $\displaystyle 0=\langle x,By\rangle_{H}=\langle
u+By,By\rangle_{H}=\langle u,By\rangle_{H}+\langle By,By\rangle_{H}$ (3.38)
$\displaystyle=\langle Au,y\rangle_{K}+\langle By,By\rangle_{H}=\langle
By,By\rangle_{H}$

implying $By=0$ so that $x=u\in\ker A$. This proves that
$(\operatorname{ran}B)^{\perp}\subseteq\ker A.$ Conversely, let $x\in\ker A$.
Then for any $y\in D(B)$ it follows by (3.31) that

(3.39) $\displaystyle 0=\langle Ax,y\rangle_{K}=\langle x,By\rangle_{H},$

which implies $x\in(\operatorname{ran}B)^{\perp}.$ This proves that $\ker
A\subseteq(\operatorname{ran}B)^{\perp}$ and thus, $\ker
A=(\operatorname{ran}B)^{\perp}$. A similar proof shows that $\ker
B=(\operatorname{ran}A)^{\perp}$. Next, it follows from the relationship $\ker
A=(\operatorname{ran}B)^{\perp}$ for the subspaces $\ker
A,\operatorname{ran}B$ that $(\ker
A)^{\perp}=(\operatorname{ran}B)^{{\perp}{\perp}}=\overline{\operatorname{ran}B}\supseteq\operatorname{ran}B$.
Conversely, if $v\in(\ker A)^{\perp}$ then by (3.32) it follows that $v=u+By$
for some $u\in\ker A,y\in D(B)$ and hence by (3.31) we have

(3.40) $\displaystyle 0=\langle u,v\rangle_{H}=\langle
u,u+By\rangle_{H}=\langle u,u\rangle_{H}+\langle u,By\rangle_{H}=\langle
u,u\rangle_{H}+\langle Au,y\rangle_{H}=\langle u,u\rangle_{H}$

implying $v=By\in\operatorname{ran}B$. This proves that $(\ker
A)^{\perp}\subseteq\operatorname{ran}B$ and thus it follows that $(\ker
A)^{\perp}=\overline{\operatorname{ran}B}=\operatorname{ran}B$. A similar
proof shows that $(\ker
B)^{\perp}=\overline{\operatorname{ran}A}=\operatorname{ran}A$. This completes
the proof of our claim.

We will now prove that $A$ and $B$ are closed operators. Suppose
$\\{x_{n}\\}_{n\in\mathbb{N}}$ is a sequence in $D(A)$ converging in $H$ to
$x$ and the sequence $\\{Ax_{n}\\}_{n\in\mathbb{N}}$ converges to $z$ in $K$.
Then, for each $y\in\ker B$, we have by (3.31) that

(3.41) $\displaystyle 0=\langle
x,By\rangle_{H}=\lim_{n\rightarrow\infty}\langle
x_{n},By\rangle_{H}=\lim_{n\rightarrow\infty}\langle
Ax_{n},y\rangle_{K}=\langle z,y\rangle_{K}$

implying $z\in(\ker B)^{\perp}$. From this and (3.36) we have $Aw=z$ for some
$w\in D(A)$. Hence, for any $y\in D(B)$, we have

(3.42) $\displaystyle\langle x,By\rangle_{H}=\lim_{n\rightarrow\infty}\langle
x_{n},By\rangle_{H}=\lim_{n\rightarrow\infty}\langle
Ax_{n},y\rangle_{K}=\langle z,y\rangle_{K}=\langle Aw,y\rangle_{K}=\langle
w,By\rangle_{K}$

implying $x-w\in(\operatorname{ran}B)^{\perp}=\ker A\subseteq D(A)$, where the
latter equality follows from (3.35). As $D(A)$ is a subspace of $H$, it
follows that $x=x-w+w\in D(A)$ and hence $Ax=Aw=z$. This proves that $A$ is a
closed operator. A similar proof shows that $B$ is a closed operator.

We will now prove that $D(A)$ and $D(B)$ are dense in $H$ and $K$,
respectively. Let $u\in D(A)^{\perp}$. Then since
$(\operatorname{ran}B)^{\perp}=\ker A\subseteq D(A)$ it follows from this and
the fact that $\operatorname{ran}B$ is closed, that
$D(A)^{\perp}\subseteq\operatorname{ran}B$. Hence, $u=By$ for some $y\in D(B)$
and for every $x\in D(A)$ we have

(3.43) $\displaystyle\langle Ax,y\rangle_{H}=\langle x,By\rangle_{H}=\langle
x,u\rangle_{H}=0$

implying $y\in(\operatorname{ran}A)^{\perp}=\ker B$ and thus, $u=By=0$. This
proves $D(A)^{\perp}=\\{0\\}$ so that
$\overline{D(A)}=D(A)^{{\perp}{\perp}}=\\{0\\}^{\perp}=H$. A similar proof
shows that $\overline{D(B)}=K$.

We have proven that $A:D(A)\rightarrow K$ and $B:D(B)\rightarrow H$ are
densely defined closed operator with closed ranges. Now by (3.31) it follows
that

(3.44) $\displaystyle D(A)\subseteq D(B^{*}),\;\;B^{*}x=Ax,\;\;\text{for all
}x\in D(A),$ (3.45) $\displaystyle D(B)\subseteq
D(A^{*}),\;\;A^{*}y=By,\;\;\text{for all }y\in D(B),$

and hence

(3.46) $\displaystyle\ker A\subseteq\ker
B^{*},\;\;\operatorname{ran}A\subseteq\operatorname{ran}B^{*},$ (3.47)
$\displaystyle\ker B\subseteq\ker
A^{*},\;\;\operatorname{ran}B\subseteq\operatorname{ran}A^{*}.$

It follows from this, (3.35), (3.36), and (by general properties of adjoints)
$\ker B\subseteq(\operatorname{ran}B^{*})^{\perp},\ker
A\subseteq(\operatorname{ran}A^{*})^{\perp}$, that

(3.48) $\displaystyle\ker B^{*}=\ker
A,\;\;\operatorname{ran}B^{*}=\operatorname{ran}A,$ (3.49) $\displaystyle\ker
A^{*}=\ker B,\;\;\operatorname{ran}A^{*}=\operatorname{ran}B.$

Therefore, by Lemma 3.10 we conclude that $B^{*}=A$ and $A^{*}=B$. This
completes the proof. ∎

##  4\. The minimal and maximal operators associated with linear differential-
algebraic equations

In this section we will consider the linear operators associated to the linear
DAEs in the following form:

(4.1) $\displaystyle J\frac{df}{dt}+Hf=Wg$

on an interval $I$ with coefficients $J,H,W$ satisfying the following:

(4.2) $\displaystyle I\text{ is an interval (open, closed, half-open, bounded,
or unbounded)},$ (4.3) $\displaystyle J\in
M_{n}(\mathbb{C}),\;\;J\not=0,\;\;J^{*}=-J,$ (4.4) $\displaystyle
H,W:I\rightarrow M_{n}(\mathbb{C}),\;\;H,W\in M_{n}(\mathcal{M}(I)),$ (4.5)
$\displaystyle W(t)^{*}=W(t)\geq 0,\;\;\det W(t)\not=0,\;\;\text{for a.e.
}t\in I.$

######  Definition 4.1.

The linear operator $\mathcal{L}$ defined on the subspace $D(\mathcal{L})$ of
$[\mathcal{M}(I)]^{n}$ by

(4.6) $\displaystyle
D(\mathcal{L})=\left\\{f\in[\mathcal{M}(I)]^{n}:Jf\in[W^{1,1}_{loc}(I)]^{n}\right\\},$
(4.7) $\displaystyle\mathcal{L}:D(\mathcal{L})\to[\mathcal{M}(I)]^{n},$ (4.8)
$\displaystyle\mathcal{L}f=W^{-1}\left(\frac{d}{dt}Jf+Hf\right),\;\;f\in
D(\mathcal{L}),$

is called the differential-algebraic (DA) operator associated with $I,J,H,W$.

The following lemma gives an alternative characterization of $D(\mathcal{L})$
and $\mathcal{L}$ in terms the orthogonal projection of $\mathbb{C}^{n}$ onto
the range of $J$. To do this we will use the Moore-Penrose pseudoinverse [15,
37, 6] of the matrix $J$.

######  Lemma 4.2.

Let $J^{+}$ denote the Moore-Penrose pseudoinverse of $J$, i.e., the unique
matrix $J^{+}\in M_{n}(\mathbb{C})$ satisfying the four (Moore-Penrose)
conditions: $JJ^{+}J=J$,
$J^{+}JJ^{+}=J^{+},(J^{+}J)^{*}=J^{+}J,(JJ^{+})^{*}=JJ^{+}$. Then
$J^{+}J=JJ^{+}$ and $JJ^{+}$ is the orthogonal projection of $\mathbb{C}^{n}$
onto $\operatorname{ran}J$. Moreover,

(4.9) $\displaystyle
D(\mathcal{L})=\left\\{f\in[\mathcal{M}(I)]^{n}:J^{+}Jf\in[W^{1,1}_{loc}(I)]^{n}\right\\},$
(4.10)
$\displaystyle\mathcal{L}f=W^{-1}\left[J\frac{d}{dt}(J^{+}Jf)+Hf\right],\;\;f\in
D(\mathcal{L})$

###### Proof.

Let $J^{+}$ denote the Moore-Penrose pseudoinverse of $J$. Then it follows
immediately from this that $J^{+}J$ and $JJ^{+}$ are the orthogonal
projections of $\mathbb{C}^{n}$ onto $\operatorname{ran}J^{*}$ and
$\operatorname{ran}J$, respectively. As $J^{*}=-J$ this implies
$\operatorname{ran}J^{*}=\operatorname{ran}J$ so that $J^{+}J=JJ^{+}$. Next,
suppose $f\in[\mathcal{M}(I)]^{n}$. Then, since $JJ^{+}J=J$ and $J$ is
independent of $t$, it follows that

(4.11) $\displaystyle Jf\in[W^{1,1}_{loc}(I)]^{n}\iff
J^{+}Jf\in[W^{1,1}_{loc}(I)]^{n}.$

The proof of the lemma now follows immediately from this. ∎

######  Example 4.3.

As an example, consider the periodic Maxwell’s DAEs (2.10). The corresponding
DA operator $\mathcal{L}$ is associated with the interval $I=\mathbb{R}$ and
the coefficients $J,H,W$ defined by (2.10c), (2.10d), (2.10e). Let $J^{+}$
denote the Moore-Penrose pseudoinverse of $J$. Then $J^{+}J=JJ^{+}$ is the
orthogonal projection of $\mathbb{C}^{6}$ onto $\operatorname{ran}J$ and, in
this example,

(4.12) $\displaystyle J^{+}=-J,\;J^{+}J=JJ^{+}=\begin{bmatrix}1&0&0&0&0&0\\\
0&1&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&0\end{bmatrix},$ (4.13) $\displaystyle
J^{+}Jf(z)=J^{+}J\begin{bmatrix}E(z)&H(z)\end{bmatrix}^{T}=\begin{bmatrix}E_{1}(z)&E_{2}(z)&0&H_{1}(z)&H_{2}(z)&0\end{bmatrix}^{T}.$

Thus, from Lemma 4.2, we see more clearly what we meant in the comment made in
Sec. 2 that it is the tangential components
$f_{1}(z)=\begin{bmatrix}E_{1}(z)&E_{2}(z)&H_{1}(z)&H_{2}(z)\end{bmatrix}^{T}$
which give rise to the differential part of the periodic Maxwell’s DAEs
(2.10). In this regard with this example in mind, see Proposition 4.17. This
concludes the example.

Next, define the Hilbert space $L^{2}(I;W)$ with inner product
$\langle\cdot,\cdot\rangle_{W}$ by

(4.14) $\displaystyle
L^{2}(I;W)=\left\\{f\in[\mathcal{M}(I)]^{n}:\int_{I}\langle
W(t)f(t),f(t)\rangle dt<\infty\right\\},$ (4.15) $\displaystyle\langle
f,g\rangle_{W}=\int_{I}\langle W(t)f(t),g(t)\rangle dt,\;\;f,g\in L^{2}(I;W),$

and the subspace $L^{2}_{loc}(I;W)$ of $\mathcal{M}(I)$ by

(4.16) $\displaystyle L^{2}_{loc}(I;W)=\\{f\in[\mathcal{M}(I)]^{n}:f\in
L^{2}([a,b];W),\text{ for every compact interval }[a,b]\subseteq I\\}.$

In particular,

(4.17) $\displaystyle L^{2}_{loc}(I;W)=L^{2}(I;W),\text{ if }I\text{ is a
compact interval.}$

######  Remark 4.4.

It easy to verify that left multiplication by $W^{-1/2}$ is an isometric
isomorphism between the Hilbert spaces $[L^{2}(I)]^{n}$ and $L^{2}(I;W)$ from
which it follows that

(4.18) $\displaystyle
L^{2}(I;W)=W^{-1/2}[L^{2}(I)]^{n},\;L^{2}_{loc}(I;W)=W^{-1/2}[L^{2}_{loc}(I)]^{n}.$

In the case that $\det J=0$, there will be an alternative characterizations of
$L^{2}(I;W)$ and $L^{2}_{loc}(I;W)$ in Lemma 4.13 that plays a key role in our
paper.

######  Definition 4.5.

The maximal operator $L:D(L)\rightarrow L^{2}(I;W)$ generated by $\mathcal{L}$
and the minimal operator $L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow
L^{2}(I;W)$ generated by $\mathcal{L}$ are defined by

(4.19) $\displaystyle D(L)=\\{f\in L^{2}(I;W):f\in
D(\mathcal{L}),\mathcal{L}f\in L^{2}(I;W)\\},$ (4.20) $\displaystyle
Lf=\mathcal{L}f,\;\;\text{for }f\in D(L),$ (4.21) $\displaystyle
D(L_{0}^{\prime})=\\{f\in D(L):Jf\text{ has compact support contained in the
interior of }I\\},$ (4.22) $\displaystyle
L_{0}^{\prime}f=\mathcal{L}f,\;\;\text{for }f\in D(L_{0}^{\prime}).$

######  Notation 4.6.

When we need to be explicit about the dependence of $\mathcal{L},L$ or
$L_{0}^{\prime}$ on $H$ and/or $J$ we will use the subscript $(\cdot)_{H}$ or
$(\cdot)_{J,H}$ with these operators, e.g., $\mathcal{L}_{H},L_{H}$ or
$(L_{0}^{\prime})_{H}$ and for the latter, we will just write
$L_{H,0}^{\prime}$ instead. Similarly, $(L_{0}^{\prime})_{J,H}$ will be
written instead as $L_{J,H,0}^{\prime}$.

As we shall see later, the following lemma and theorem are fundamental in the
developing the spectral theory associated with the operator $L$ on the Hilbert
space $L^{2}(I;W)$.

######  Lemma 4.7.

The sets $D(L_{0}^{\prime})$ and $D(L)$ are subspaces of the Hilbert space
$L^{2}(I;W)$. Moreover, $L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow
L^{2}(I;W)$ and $L:D(L)\rightarrow L^{2}(I;W)$ are linear operators with

(4.23) $\displaystyle D(L_{0}^{\prime})\subseteq D(L)\subseteq
D(\mathcal{L})\cap L^{2}(I;W),$ (4.24) $\displaystyle
L_{0}^{\prime}f=Lf=\mathcal{L}f,\;\;f\in D(L_{0}^{\prime}).$

###### Proof.

The proof is straightforward and so will be omitted. ∎

######  Remark 4.8.

It follows from (3.20) and (3.21) that the linear operators
$\mathcal{L}_{H^{*}},L_{H^{*}},$ and $L_{H^{*},0}^{\prime}$ can be defined as
above in terms of $H^{*}$ instead of $H$ and, in particular, Lemma 4.7 is true
too for these operators.

######  Theorem 4.9.

a) For $f\in D(L_{0}^{\prime})$ and $g\in D(L_{H^{*}})$ we have

(4.25) $\displaystyle\langle L_{0}^{\prime}f,g\rangle_{W}=\langle
f,L_{H^{*}}g\rangle_{W}.$

b) For $f\in D(L_{0}^{\prime})$ and $g\in D(L_{H^{*},0}^{\prime})$ we have

(4.26) $\displaystyle\langle L_{0}^{\prime}f,g\rangle_{W}=\langle
f,L_{H^{*},0}^{\prime}g\rangle_{W}.$

###### Proof.

a): Let $f\in D(L_{0}^{\prime})$. Then there exists a compact interval
$[a,b]\subseteq I$ such that $(Jf)(t)=0$ for every $t\in I\setminus(a,b)$ and,
in particular, $(Jf)(a)=(Jf)(b)=0$. Hence, for any $g\in D(L_{H^{*}})$ and by
Lemma 4.2, we have

$\displaystyle\langle L_{0}^{\prime}f,g\rangle_{W}$
$\displaystyle=\langle\mathcal{L}f,g\rangle_{W}$
$\displaystyle=\int_{I}\left\langle\left[\frac{d}{dt}Jf(t)+H(t)f(t)\right],g(t)\right\rangle
dt$
$\displaystyle=\int_{I}\left\langle\frac{d}{dt}Jf(t),J^{+}Jg(t)\right\rangle+\left\langle
H(t)f(t),g(t)\right\rangle dt$ $\displaystyle=\int_{I}\frac{d}{dt}\left\langle
Jf(t),J^{+}Jg\right\rangle-\left\langle
Jf(t),\frac{d}{dt}J^{+}Jg(t)\right\rangle+\left\langle
f(t),H(t)^{*}g(t)\right\rangle dt$
$\displaystyle=\int_{I}\frac{d}{dt}\left\langle
Jf(t),J^{+}Jg(t)\right\rangle+\left\langle
W(t)f(t),W(t)^{-1}\left[\frac{d}{dt}Jg(t)+H(t)^{*}g(t)\right]\right\rangle dt$
$\displaystyle=\int_{I}\frac{d}{dt}\left\langle Jf(t),J^{+}Jg(t)\right\rangle
dt+\langle f,L_{H^{*}}g\rangle_{W}$
$\displaystyle=\left\langle(Jf)(b),J^{+}(Jg)(b)\right\rangle-\left\langle(Jf)(a),J^{+}(Jg)(a)\right\rangle+\langle
f,L_{H^{*}}g\rangle_{W}$ $\displaystyle=\langle f,L_{H^{*}}g\rangle_{W}.$

b): This follows immediately from part a) since
$D(L_{H^{*},0}^{\prime})\subseteq D(L_{H^{*}})$ and
$L_{H^{*},0}^{\prime}g=L_{H_{*}}g$ if $g\in D(L_{H^{*},0}^{\prime})$. ∎

Consider the case that $\det J=0$. Let $V\in M_{n}(\mathbb{C})$ be any unitary
matrix, i.e.,

(4.27) $\displaystyle V\in M_{n}(\mathbb{C}),\;\;V^{*}=V^{-1},$

such that $V^{-1}JV=[J_{ij}]_{i,j=1,2}$ has the $2\times 2$ block partitioned
matrix form

(4.28) $\displaystyle
V^{-1}JV=[J_{ij}]_{i,j=1,2}=\begin{bmatrix}J_{11}&J_{12}\\\ J_{21}&J_{22}\\\
\end{bmatrix}=\begin{bmatrix}J_{11}&0\\\ 0&0\\\
\end{bmatrix},\quad\det(J_{11})\neq 0,$

and, more precisely,

(4.29) $\displaystyle J_{ij}\in M_{n_{i}\times
n_{j}}(\mathbb{C}),i,j=1,2;\;\;J_{ij}=0,\;\;(i,j)\not=(1,1),\;\;\det(J_{11})\neq
0,$

where we define $n_{1},n_{2}\in\mathbb{N}$ by

(4.30) $\displaystyle n_{1}:=\operatorname{rank}J,\;\;n_{2}:=\dim\ker
J=\operatorname{nullity}(J)=n-n_{1}.$

######  Remark 4.10.

It should be noted that each and every such $V$ can be constructed in the
following manner: Let $v_{1},\ldots,v_{n_{1}}$ be an orthonormal basis for
$\operatorname{ran}J$ and $v_{n_{1}+1},\ldots,v_{n_{2}}$ an orthonormal basis
for $\operatorname{ker}J$. Then the $n\times n$ column matrix
$V=\begin{bmatrix}v_{1}|\cdots|v_{n_{1}}|v_{n_{1}+1}|\cdots|v_{n_{2}}\end{bmatrix}$
satisfies (4.27) and (4.28). The key point in this remark is the converse of
this statement is also true and this gives insight into our reason for
introducing such a unitary matrix $V$ above.

Now block partition the matrices $V^{-1}HV=[H_{ij}]_{i,j=1,2}$ and
$V^{-1}WV=[W_{ij}]_{i,j=1,2}$ conformal to the block structure of $V^{-1}JV$
in (4.28),

(4.31) $\displaystyle
V^{-1}HV=[H_{ij}]_{i,j=1,2}=\begin{bmatrix}H_{11}&H_{12}\\\
H_{21}&H_{22}\end{bmatrix}\in M_{n}(\mathcal{M}(I)),$ (4.32) $\displaystyle
V^{-1}WV=[W_{ij}]_{i,j=1,2}=\begin{bmatrix}W_{11}&W_{12}\\\
W_{21}&W_{22}\end{bmatrix}\in M_{n}(\mathcal{M}(I)),$

and, more precisely,

(4.33) $\displaystyle H_{ij},W_{ij}\in M_{n_{i}\times
n_{j}}(\mathcal{M}(I)),\;\;i,j=1,2.$

If, in addition, $H_{22}^{-1}\in M_{n_{2}}(\mathcal{M}(I))$ [equivalently,
$\det H_{22}(t)\not=0$ for a.e. $t\in I$] then, for the Schur complement (see
[64]) of $V^{-1}HV$ with respect to $H_{22}$, we have

(4.34) $\displaystyle H/H_{22}:=H_{11}-H_{21}H_{22}^{-1}H_{12}\in
M_{n_{1}}(\mathcal{M}(I)).$

######  Lemma 4.11.

Suppose $W$ satisfies the hypotheses (4.4) and (4.5). Then the inverse
$W(t)^{-1}$ of $W(t)$ exists for a.e. $t\in I$ (and setting it to say the
identity matrix $I_{n}$ when it doesn’t exist), defines the function
$W^{-1}:I\rightarrow M_{n}(\mathbb{C})$ which satisfies $W^{-1}(t)=W(t)^{-1}$
for a.e. $t\in I$. In addition, $W^{-1}$ satisfies the same hypotheses (4.4)
and (4.5) as $W$, i.e.,

(4.35) $\displaystyle W^{-1}:I\rightarrow M_{n}(\mathbb{C}),\;\;W^{-1}\in
M_{n}(\mathcal{M}(I)),$ (4.36) $\displaystyle W^{-1}(t)^{*}=W^{-1}(t)\geq
0,\;\;\det W^{-1}(t)\not=0,\;\;\text{for a.e.\ }t\in I.$

Furthermore, the positive square root $W(t)^{\pm 1/2}$ of $W^{\pm 1}(t)$
exists for a.e. $t\in I$ (and setting it to say the identity matrix $I_{n}$
when it doesn’t exist), defines the function $W^{\pm 1/2}:I\rightarrow
M_{n}(\mathbb{C})$ which satisfies $W^{\pm 1/2}(t)=W(t)^{\pm 1/2}$ for a.e.
$t\in I$. Moreover, $W^{\pm 1/2}$ satisfies the same hypotheses (4.4) and
(4.5) as $W$, i.e.,

(4.37) $\displaystyle W^{\pm 1/2}:I\rightarrow M_{n}(\mathbb{C}),\;\;W^{\pm
1/2}\in M_{n}(\mathcal{M}(I)),$ (4.38) $\displaystyle W^{\pm
1/2}(t)^{*}=W^{\pm 1/2}(t)\geq 0,\;\;\det W^{\pm 1/2}(t)\not=0,\;\;\text{for
a.e.\ }t\in I.$

###### Proof.

The proof is obvious from the elementary theory of matrices and that for
positive definite matrices. ∎

######  Lemma 4.12.

Let $W$ satisfy the hypotheses (4.4) and (4.5) and $V$ satisfy the hypotheses
(4.27). Then, with respect to the $2\times 2$ block matrix partitioning
$V^{-1}WV=[W_{i,j}]_{i,j=1,2}$ as in (4.32), we have $W_{11}^{-1}\in
M_{n_{1}}(\mathcal{M}(I))$ and $W_{22}^{-1}\in M_{n_{2}}(\mathcal{M}(I))$, and
the Schur complements of $V^{-1}WV$ with respect to $W_{11}$ and $W_{22}$,
i.e.,

(4.39) $\displaystyle W/W_{11}:=W_{22}-W_{21}W_{11}^{-1}W_{12},$ (4.40)
$\displaystyle W/W_{22}:=W_{11}-W_{12}W_{22}^{-1}W_{21},$

respectively, have the following properties for $i=1,2$:

(4.41) $\displaystyle W/W_{ii}\in M_{n_{i}}(\mathcal{M}(I)),$ (4.42)
$\displaystyle(W/W_{ii})(t)^{*}=(W/W_{ii})(t)\geq
0,\;\;\det(W/W_{ii})(t)\not=0,\;\;\text{for a.e.\ }t\in I,$ (4.43)
$\displaystyle 0\leq(W/W_{ii})(t)\leq W_{jj}(t)\;\;\text{for a.e.\ }t\in
I,\text{ for each }j=1,2,\;j\not=i.$

###### Proof.

The proof is immediate from elementary properties of $2\times 2$ block
partitioned positive definite matrices and their Schur complements (see, for
instance, [64]) since $W(t)^{*}=W(t)\geq 0$ and $\det W(t)\not=0$ for a.e.
$t\in I$ implies $W(t)$ is a positive definite matrix for a.e. $t\in I$ which
implies so is $V^{-1}W(t)V=[W_{i,j}(t)]_{i,j=1,2}$ and hence so are the blocks
$W_{11}(t)$ and $W_{22}(t)$ with $W_{12}(t)^{*}=W_{21}(t)$. The results now
follows immediately from this. ∎

We will now characterize, in the case $\det J=0$, the spaces $L^{2}(I;W)$ and
$L^{2}_{loc}(I;W)$.

######  Lemma 4.13.

Suppose $I,W$ satisfy (4.2), (4.4), and (4.5), $V$ satisfy the hypotheses
(4.27), and block matrix partition $V^{-1}WV=[W_{i,j}]_{i,j=1,2}$ as in
(4.32). Then

(4.44) $\displaystyle f\in(\mathcal{M}(I))^{n}\iff
V^{-1}f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},\;\;f_{i}\in(\mathcal{M}(I))^{n_{i}},\;\;i=1,2.$

Furthermore,

(4.45) $\displaystyle f\in L^{2}(I;W)$ $\displaystyle\iff$ (4.46)
$\displaystyle(W/W_{11})^{1/2}f_{2}\in(L^{2}(I))^{n_{2}},\;\;W_{11}^{-1/2}(W_{11}f_{1}+W_{12}f_{2})\in(L^{2}(I))^{n_{1}}$
$\displaystyle\iff$ (4.47)
$\displaystyle(W/W_{22})^{1/2}f_{1}\in(L^{2}(I))^{n_{1}},\;\;W_{22}^{-1/2}(W_{21}f_{1}+W_{22}f_{2})\in(L^{2}(I))^{n_{2}}.$

Similarly, the statement remains true if we replace
$L^{2}(I;W),(L^{2}(I))^{n_{1}},(L^{2}(I))^{n_{2}}$ in (4.45), (4.46), (4.47)
by $L^{2}_{loc}(I;W),(L^{2}_{loc}(I))^{n_{1}},(L^{2}_{loc}(I))^{n_{2}},$
respectively.

###### Proof.

The proof of the statement (4.44) is obvious and so will be omitted. Also,
once we proof the equivalence of the statements (4.45), (4.46), and (4.47)
then the equivalence of these statements in the local, i.e., “loc,” case
follows immediately. Now we prove that (4.45) iff (4.47), but we will omit the
proof of (4.45) iff (4.46) as it is similar. By Lemma 4.11, we have the block
factorization

(4.48) $\displaystyle
V^{-1}WV=[W_{ij}]_{i,j=1,2}=\begin{bmatrix}I_{n_{1}}&W_{12}W_{22}^{-1}\\\
0&I_{n_{2}}\end{bmatrix}\begin{bmatrix}W/W_{22}&0\\\
0&W_{22}\end{bmatrix}\begin{bmatrix}I_{n_{1}}&0\\\
W_{22}^{-1}W_{21}&I_{n_{2}}\end{bmatrix}.$

Next, it follows from this and Lemma 4.12 together with Lemma 4.11 applied to
$W/W_{22}$ that for any $f\in(\mathcal{M}(I))^{n}$ we have

(4.49)
$\displaystyle\int_{I}(W(t)f(t),f(t))dt=\int_{I}(V^{-1}W(t)V[V^{-1}f(t)],[V^{-1}f(t)])dt$
(4.50) $\displaystyle=\int_{I}\left(\begin{bmatrix}W/W_{22}&0\\\
0&W_{22}\end{bmatrix}\begin{bmatrix}I_{n_{1}}&0\\\
W_{22}^{-1}W_{21}&I_{n_{2}}\end{bmatrix}\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},\begin{bmatrix}I_{n_{1}}&0\\\
W_{22}^{-1}W_{21}&I_{n_{2}}\end{bmatrix}\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix}\right)(t)dt$ (4.51)
$\displaystyle\int_{I}((W/W_{22})(t)f_{1}(t),f_{2}(t))+(W_{22}(t)[W_{22}^{-1}(t)W_{21}(t)f_{1}(t)+f_{2}],W_{21}(t)f_{1}(t)+f_{2}(t))dt$
(4.52)
$\displaystyle=\int_{I}||[(W/W_{22})^{1/2}f_{1}](t)||^{2}dt+\int_{I}||[W_{22}^{-1/2}(W_{21}f_{1}+W_{22}f_{2}](t))||^{2}dt.$

The proof that (4.45) iff (4.47) follows immediately from this, which
completes the proof. ∎

######  Definition 4.14 (Index-1 hypotheses).

The following set of hypotheses are called the local index-1 hypotheses for
$H,W$ with respect to $J$ on the interval $I$:

(4.53) $\displaystyle\text{If }\det J\not=0\text{ then }H,W\in
M_{n}(L^{1}_{loc}(I)).$

If $\det J=0$ then

(4.54) $\displaystyle H_{22}^{-1}\in M_{n_{2}}(\mathcal{M}(I)),$ (4.55)
$\displaystyle H_{12}H_{22}^{-1}W_{22}^{1/2}\in M_{n_{1}\times
n_{2}}(L^{2}_{loc}(I)),$ (4.56) $\displaystyle H/H_{22},W_{11}\in
M_{n_{1}}(L^{1}_{loc}(I)),$ (4.57) $\displaystyle
W_{22}^{1/2}H_{22}^{-1}W_{22}^{1/2}\in M_{n_{2}}(L^{\infty}_{loc}(I)),$ (4.58)
$\displaystyle W_{22}^{1/2}(W_{22}^{-1}W_{21}-H_{22}^{-1}H_{21})\in
M_{n_{2}\times n_{1}}(L^{2}_{loc}(I)),$ (4.59) $\displaystyle W/W_{22}\in
M_{n_{1}}(L^{1}_{loc}(I))$

Similarly, if we drop in the hypotheses above the “loc” then these are called
the index-1 hypotheses.

The next lemma and corollary gives useful simplification of these hypotheses.

######  Lemma 4.15.

Suppose $\det J=0$. Then the local index-$1$ hypotheses for $H,W$ with respect
to $J$ on the interval $I$ are true if and only if the following conditions
are satisfied:

(4.60) $\displaystyle H_{22}^{-1}\in
M_{n_{2}}(\mathcal{M}(I)),\;W_{22}^{1/2}H_{22}^{-1}W_{22}^{1/2}\in
M_{n_{2}}(L^{\infty}_{loc}(I))$ (4.61) $\displaystyle H/H_{22},W_{11}\in
M_{n_{1}}(L^{1}_{loc}(I)),$ (4.62) $\displaystyle
H_{12}H_{22}^{-1}W_{22}^{1/2}\in M_{n_{1}\times
n_{2}}(L^{2}_{loc}(I)),\;W_{22}^{1/2}H_{22}^{-1}H_{21}\in M_{n_{2}\times
n_{1}}(L^{2}_{loc}(I)).$

Similarly, the statement with the “local” and “loc” dropped is true.

###### Proof.

Suppose $\det J=0$. Then by hypotheses on $W$ we have

(4.63) $\displaystyle
W_{11}-W/W_{22}=W_{12}W_{22}^{-1}W_{21}=(W_{22}^{-1/2}W_{21})^{*}(W_{22}^{-1/2}W_{21})$

and $0\leq W/W_{22}\leq W_{11}$. Assume that $W_{11}\in
M_{n_{1}}(L^{1}_{loc}(I))$. Then this implies $W/W_{22}\in
M_{n_{1}}(L^{1}_{loc}(I))$ and $W_{22}^{-1/2}W_{21}\in M_{n_{2}\times
n_{1}}(L^{2}_{loc}(I))$. And from this it follows that

(4.64) $\displaystyle\;W_{22}^{1/2}(W_{22}^{-1}W_{21}-H_{22}^{-1}H_{21})\in
M_{n_{2}\times n_{1}}(L^{2}_{loc}(I))$ (4.65) $\displaystyle\iff
W_{22}^{1/2}H_{22}^{-1}H_{21}\in M_{n_{2}\times n_{1}}(L^{2}_{loc}(I)).$

The proof of the lemma now follows immediately from this. ∎

######  Corollary 4.16.

The (local) index-$1$ hypotheses for $H,W$ with respect to $J$ on the interval
$I$ are true if and only if the (local) index-$1$ hypotheses for $H^{*},W$
with respect to $J$ on the interval $I$ are true.

###### Proof.

The proof of this corollary follows immediately from considering matrix
adjoints of the conditions in Lemma 4.15 if $\det J=0$ or those in Def. 4.14
if $\det J\not=0.$ ∎

The reason for such hypotheses will become clear as we move forward in this
section, but the next proposition gives the main reason why.

######  Proposition 4.17.

Suppose $I,J,H,W$ satisfy (4.2)–(4.5) and, in the case $\det J=0$, that $V$
satisfies the hypotheses (4.27), $V^{-1}JV=[W_{i,j}]_{i,j=1,2}$ has the block
form (4.28), and $V^{-1}HV=[H_{i,j}]_{i,j=1,2},V^{-1}WV=[W_{i,j}]_{i,j=1,2}$
have the conformal block structure in (4.31), (4.32), respectively. Then the
following statements are true:
(i) If (4.54) then

(4.66) $\displaystyle\mathcal{L}f=g$ $\displaystyle\iff$ (4.67) $\displaystyle
g\in[\mathcal{M}(I)]^{n},$ (4.68) $\displaystyle
f_{1}\in[W^{1,1}_{loc}(I)]^{n_{1}},$ (4.69) $\displaystyle
J_{11}\frac{df_{1}}{dt}+H/H_{22}f_{1}=F,$ (4.70) $\displaystyle
f_{2}=H_{22}^{-1}(W_{21}g_{1}+W_{22}g_{2})-H_{22}^{-1}H_{21}f_{1},$

where $F\in[\mathcal{M}(I)]^{n_{2}}$ is defined by

(4.71) $\displaystyle
F=(W_{11}g_{1}+W_{12}g_{2})-H_{12}H_{22}^{-1}(W_{21}g_{1}+W_{22}g_{2}),$

and $f,g,f_{1},f_{2},g_{1},g_{2}$ are related by

(4.72) $\displaystyle V^{-1}f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},\;\;V^{-1}g=\begin{bmatrix}g_{1}\\\ g_{2}\end{bmatrix}.$

(ii) If (4.53) or (4.54)–(4.56) then for any $g\in L^{2}_{loc}(I;W),$
$t_{0}\in I,$ and $f_{0}\in\operatorname{ran}J$ there is a unique solution
$f\in D(\mathcal{L})$ to the IVP

(4.73) $\displaystyle\mathcal{L}f=g,\;\;(Jf)(t_{0})=f_{0}.$

In addition, in the case $\det J\not=0$ or if (4.57)–(4.59) in the case $\det
J=0$, then

(4.74) $\displaystyle f\in L^{2}_{loc}(I;W).$

###### Proof.

(i): Assume the hypotheses. ($\Rightarrow$): Suppose (4.66) is true. Then
$f\in D(L)$ and so $Jf\in[W^{1,1}_{loc}(I)]^{n}$. This implies that
$f_{1}\in[W^{1,1}_{loc}(I)]^{n_{1}},f_{2}\in[\mathcal{M}(I)]^{n_{2}},$ where
$f_{1},f_{2}$ are related to $f$ by (4.72). Next, since $\mathcal{L}f=g$ then
$g\in[\mathcal{M}(I)]^{n}$ and $\frac{d}{dt}(Jf)+Hf=W\mathcal{L}f=Wg$ (with
equality in $[\mathcal{M}(I)]^{n}$) which yields the system of equations:

(4.75) $\displaystyle
J_{11}\frac{df_{1}}{dt}+H_{11}f_{1}+H_{12}f_{2}=W_{11}g_{1}+W_{12}g_{2},$
(4.76) $\displaystyle H_{21}f_{1}+H_{22}f_{2}=W_{21}g_{1}+W_{22}g_{2},$

where $g_{1},g_{2}$ are related to $f$ by (4.72). Solving these equations for
$f_{2}$ in terms of $f_{1},g_{1},g_{2}$ yields the equivalent system of
equations:

(4.77) $\displaystyle
J_{11}\frac{df_{1}}{dt}+H/H_{22}f_{1}=W_{11}g_{1}+W_{12}g_{2}-H_{12}H_{22}^{-1}(W_{21}g_{1}+W_{22}g_{2}),$
(4.78) $\displaystyle
f_{2}=H_{22}^{-1}(W_{21}g_{1}+W_{22}g_{2})-H_{22}^{-1}H_{21}f_{1},$

which are equivalent to the system of equations (4.69) and (4.70), where $F$
is defined by (4.71). Thus, we have proven that (4.66) implies (4.67)–(4.71),
with $f,g$ and $f_{1},f_{2},g_{1},g_{2}$ are related by (4.72).
($\Leftarrow$): Conversely, suppose $f,g$ satisfy (4.67)–(4.71), where $f,g$
and $f_{1},f_{2},g_{1},g_{2}$ are related by (4.72). Then one can verify that
$f\in D(\mathcal{L})$ and $\frac{d}{dt}(Jf)+Hf=Wg$ (with equality in
$[\mathcal{M}(I)]^{n}$), which implies $\mathcal{L}f=g$. This completes the
proof of (i). (ii): Assume (4.53), in the case $\det J\not=0$, or, in the case
$\det J=0$, that (4.54)–(4.56). Let $g\in L^{2}_{loc}(I;W),t_{0}\in
I,f_{0}\in\operatorname{ran}J$.

Consider the first case in which $\det J\not=0$ and hypotheses (4.53) are
true. Then $Wg\in[L^{1}_{loc}(I)]^{n}$ since, for any compact interval
$[a,b]\subseteq I$, we have $g\in L^{2}([a,b];W)$ and $W\in
M_{n}(L^{1}([a,b]))$ by hypothesis so it follows by Holder’s inequality that

(4.79)
$\displaystyle\int_{[a,b]}||W(t)g(t)||dt=\int_{[a,b]}(W(t)g(t),W(t)g(t))^{1/2}dt$
(4.80) $\displaystyle\leq\int_{[a,b]}||W(t)||^{1/2}||W(t)^{1/2}g(t)||dt$
(4.81)
$\displaystyle\leq\int_{[a,b]}(||W(t)||^{1/2})^{2}dt^{1/2}\int_{[a,b]}||W(t)^{1/2}g(t)||^{2}dt^{1/2}$
(4.82)
$\displaystyle=\left[\int_{[a,b]}||W(t)||dt\int_{[a,b]}(W(t)g(t),g(t))dt\right]^{1/2}<\infty.$

It also follows from this that $J^{-1}Wg\in[L^{1}_{loc}(I)]^{n}$ and by
hypotheses that $J^{-1}H\in M_{n}(L^{1}_{loc}(I))$. By Theorem 3.1, there
exists a unique solution $f\in[W^{1,1}_{loc}(I)]^{n}=D(\mathcal{L})$ to the
ODE IVP

(4.83) $\displaystyle\frac{d}{dt}f+J^{-1}Hf=J^{-1}Wg,\;\;f(t_{0})=J^{-1}f_{0}$

on $I$. The proof of the statement (ii) in the case $\det J\not=0$ now follows
immediately from this with the exception that we still need to prove $f\in
L^{2}_{loc}(I;W)$. But this follows from the fact that
$f\in[W^{1,1}_{loc}(I)]^{n}\subseteq[L^{\infty}_{loc}(I)]^{n}$ so by Holder’s
inequality we have for any compact interval $[a,b]\subseteq I$,

(4.84)
$\displaystyle\int_{[a,b]}(W(t)f(t),f(t))dt\leq\int_{[a,b]}||W(t)||||f(t)||^{2}dt$
(4.85) $\displaystyle\leq\left(\operatorname*{ess\,sup}_{t\in
I}||f(t)||\right)^{2}\int_{[a,b]}||W(t)||dt<\infty,$

implying that $f\in L^{2}_{loc}(I;W)$.

Consider now, the second case in which $\det J=0$ and hypotheses (4.54)–(4.56)
are true. First, it follows from the block structure (4.28) of $V^{-1}JV$ that

(4.86) $\displaystyle V^{-1}f_{0}=\begin{bmatrix}(f_{1})_{0}\\\
0\end{bmatrix}$

for a unique $(f_{1})_{0}\in\mathbb{C}^{n_{1}}$. Next, by hypothesis (4.56) we
know that $W_{11}\in M_{n_{1}}(L^{1}_{loc}(I))$ from which it follows that
$W_{11}^{1/2}\in M_{n_{1}}(L^{2}_{loc}(I))$. This together with $g\in
L^{2}_{loc}(I;W)$, Lemma 4.13, and hypothesis (4.55) implies by Holder’s
inequality that $F$ defined in (4.71) satisfies

(4.87) $\displaystyle
F=W_{11}^{1/2}[W_{11}^{-1/2}(W_{11}g_{1}+W_{12}g_{2})]-H_{12}H_{22}^{-1}W_{22}^{1/2}[W_{22}^{-1/2}(W_{21}g_{1}+W_{22}g_{2})],$
(4.88) $\displaystyle F\in[L^{1}_{loc}(I)]^{n_{1}}.$

This and hypothesis (4.56) implies that
$J_{11}^{-1}F\in[L^{1}_{loc}(I)]^{n_{1}}$ and $J_{11}^{-1}H/H_{22}\in
M_{n_{1}}(L^{1}_{loc}(I))$. By Theorem 3.1, there exists a unique solution
$f_{1}\in[W^{1,1}_{loc}(I)]^{n_{1}}$ to the ODE IVP

(4.89)
$\displaystyle\frac{df_{1}}{dt}+J_{11}^{-1}H/H_{22}f_{1}=J_{11}^{-1}F,\;\;f_{1}(t_{0})=J_{11}^{-1}(f_{1})_{0}.$

on $I$. Thus, if we take $f_{2}$ to be defined by (4.70) then $f,g$ satisfy
(4.67)–(4.71), where $f,g$ and $f_{1},f_{2},g_{1},g_{2}$ are related by
(4.72). It follows from this and part (i) of this proposition, that $f\in
D(\mathcal{L}),$ $\mathcal{L}f=g$, and we have

(4.90)
$\displaystyle(Jf)(t_{0})=(JVV^{-1}f)(t_{0})=V\begin{bmatrix}J_{11}f_{1}(t_{0})\\\
0\end{bmatrix}=V\begin{bmatrix}(f_{1})_{0}\\\ 0\end{bmatrix}=f_{0}.$

The uniqueness portion of statement (ii) in this case follows immediately by
the uniqueness of the solution to the ODE IVP (4.89) on $I$. This proves
statement (ii) in the case $\det J=0$ with the exception that we still need to
prove $f\in L^{2}_{loc}(I;W)$ if (4.57)–(4.59). We do this next.

Suppose, in addition, that (4.57)–(4.59) are true. To prove $f\in
L^{2}_{loc}(I;W)$ it suffices by Lemma 4.13 to prove $f_{1},f_{2}$ satisfy
(4.47) in the “loc” case, i.e., when in (4.47) we replace
$[L^{2}(I)]^{n_{1}},[L^{2}(I)]^{n_{1}}$ with
$[L^{2}_{loc}(I)]^{n_{1}},[L^{2}_{loc}(I)]^{n_{1}}$, respectively. Let
$[a,b]\subseteq I$ be a compact interval. First, by the hypothesis (4.59) we
have $W/W_{22}\in M_{n_{1}}(L^{1}([a,b]))$, and so by Lemma 4.12 and then
Lemma 4.11 applied to $W/W_{22}$ (instead of $W$) it follows that
$(W/W_{22})^{1/2}\in M_{n_{1}}(L^{2}([a,b]))$. Hence, since
$f_{1}\in[W^{1,1}([a,b])]^{n_{1}}\subseteq[L^{\infty}([a,b])]^{n_{1}}$, this
implies by Holder’s inequality that
$(W/W_{22})^{1/2}f_{1}\in[L^{2}([a,b])]^{n_{1}}$. Also, as
$f_{1}\in[L^{\infty}(I)]^{n_{1}}$ and by hypothesis (4.58), it follows by
Holder’s inequality that

(4.91) $\displaystyle
W_{22}^{1/2}(W_{22}^{-1}W_{21}-H_{22}^{-1}H_{21})f_{1}\in(L^{2}([a,b]))^{n_{2}}.$

Next, as $g\in L^{2}([a,b];W)$, it follows by Lemma 4.13 that

(4.92) $\displaystyle
W_{22}^{-1/2}(W_{21}g_{1}+W_{22}g_{2})\in(L^{2}([a,b]))^{n_{2}}.$

Hence, from this and hypothesis (4.57), it follows by Holder’s inequality that

(4.93) $\displaystyle
W_{22}^{1/2}H_{22}^{-1}W_{22}^{1/2}[W_{22}^{-1/2}(W_{21}g_{1}+W_{22}g_{2})]\in(L^{2}([a,b]))^{n_{2}}.$

Finally, it follows from these facts, the formula (4.70) for $f_{2}$, and
Minkowski’s inequality that

(4.94) $\displaystyle W_{22}^{-1/2}(W_{21}f_{1}+W_{22}f_{2})$ (4.95)
$\displaystyle=W_{22}^{-1/2}(W_{21}f_{1}+W_{22}(H_{22}^{-1}(W_{21}g_{1}+W_{22}g_{2})-H_{22}^{-1}H_{21}f_{1})))$
(4.96)
$\displaystyle=W_{22}^{-1/2}(W_{21}f_{1}-W_{22}H_{22}^{-1}H_{21}f_{1})+W_{22}^{1/2}H_{22}^{-1}(W_{21}g_{1}+W_{22}g_{2})$
(4.97) $\displaystyle=W_{22}^{1/2}(W_{22}^{-1}W_{21}-H_{22}^{-1}H_{21})f_{1}$
(4.98)
$\displaystyle+W_{22}^{1/2}H_{22}^{-1}W_{22}^{1/2}[W_{22}^{-1/2}(W_{21}g_{1}+W_{22}g_{2})]\in(L^{2}([a,b]))^{n_{2}}.$

This proves that $f\in L^{2}([a,b];W)$. As $[a,b]$ was an arbitrary compact
interval in $I$, this proves $f\in L^{2}_{loc}(I;W)$ which completes the proof
of statement (ii) and hence proves the proposition. ∎

######  Corollary 4.18.

If the index-$1$ hypotheses (4.53) or (4.54)–(4.59) are true on a compact
interval $I$ then

(4.99) $\displaystyle\operatorname{ran}L=L^{2}(I;W),\;\;\ker
L=\ker\mathcal{L},\;\;\dim\ker L=\operatorname{rank}J<\infty.$

###### Proof.

Let $g\in L^{2}(I;W)$, where $I$ is a compact interval. Fix any $t_{0}\in I$
and $f_{0}\in\operatorname{ran}J$. Then by Proposition 4.17 there exists a
unique $f\in D(\mathcal{L})$ such that $\mathcal{L}f=g$ with
$(Jf)(t_{0})=f_{0}$. Moreover, from this proposition we also know that $f\in
L^{2}_{loc}(I;W)=L^{2}(I;W)$. It follows from these facts that $f\in D(L)$ and
$Lf=\mathcal{L}f=g$ which implies $f\in\operatorname{ran}L.$ This proves that
$L^{2}(I;W)\subseteq\operatorname{ran}L$ and since
$\operatorname{ran}L\subseteq L^{2}(I;W)$ (by definition of $L$), we conclude
that $\operatorname{ran}L=L^{2}(I;W)$. Next, as we can take $g=0$ in this
proof, it follows immediately that $\ker L=\ker\mathcal{L}$. It remains to
prove $\dim\ker L=\operatorname{rank}J<\infty$. First, as
$J\in\mathbb{C}^{n},J\not=0$, then $1\leq
r:=\operatorname{rank}J=\dim\operatorname{ran}J\leq n<\infty$. Next, let
$\beta_{1},\ldots,\beta_{r}$ be a basis for $\operatorname{ran}J$. Then in our
proof with $g=0$ and $t_{0}\in I$ fixed, there exists a unique solution
$f_{j}\in D(L)$ to $Lf_{j}=0$ with $(Jf_{j})(t_{0})=\beta_{j}$, for each
$j=1,\ldots,r$. We claim that $f_{1},\ldots,f_{r}$ is a basis for $\ker L$.
Obviously, $f_{1},\ldots,f_{r}\in\ker L$. Next, let
$c_{1},\ldots,c_{r}\in\mathbb{C}$ be such that

(4.100) $\displaystyle c_{1}f_{1}+\cdots+c_{r}f_{r}=0.$

Then

(4.101) $\displaystyle
0=(J0)(t_{0})=[J(c_{1}f_{1}+\cdots+c_{r}f_{r})](t_{0})=c_{1}\beta_{1}\cdots+c_{r}\beta_{r}$

implying $c_{1}=\cdots=c_{r}$. This proves the vectors $f_{1},\ldots,f_{r}$
are linearly independent. Finally, let $h\in\ker L$. Then there exists scalars
$c_{1},\ldots,c_{r}\in\mathbb{C}$ such that

(4.102)
$\displaystyle(Jh)(t_{0})=c_{1}\beta_{1}\cdots+c_{r}\beta_{r}=[J(c_{1}f_{1}+\cdots+c_{r}f_{r})](t_{0}).$

By the uniqueness of the solution $f\in D(L)$ to
$Lf=0,(Jf)(t_{0})=(Jh)(t_{0})$, it follows that
$h=c_{1}f_{1}+\cdots+c_{r}f_{r}$. This proves that the vectors
$f_{1},\ldots,f_{r}$ span $\ker L$. Therefore, the vectors
$f_{1},\ldots,f_{r}$ are a basis for $\ker L$ which proves our claim and also
proves that $\dim\ker L=r=\operatorname{rank}J<\infty$. ∎

######  Definition 4.19.

In the case that $I$ is a bounded interval with endpoints $a,b$ and $a<b$, the
closed minimal operator $L_{0}:D(L_{0})\rightarrow L^{2}(I;W)$ generated by
$\mathcal{L}$ is defined by

(4.103) $\displaystyle D(L_{0})$ $\displaystyle=\\{f\in
D(L):\lim_{t_{0}\rightarrow a^{+}}(Jf)(t_{0})=\lim_{t_{1}\rightarrow
b^{-}}(Jf)(t_{1})=0\\},$ (4.104) $\displaystyle L_{0}f$
$\displaystyle=\mathcal{L}f,\text{ for }f\in D(L_{0}).$

######  Lemma 4.20.

Suppose $I$ is a bounded interval. Then $D(L_{0})$ is a subspace of the
Hilbert space $L^{2}(I;W)$. Moreover, $L_{0}:D(L_{0})\rightarrow L^{2}(I;W)$
is a linear operator with

(4.105) $\displaystyle D(L_{0}^{\prime})\subseteq D(L_{0})\subseteq
D(L)\subseteq D(\mathcal{L})\cap L^{2}(I;W),$ (4.106) $\displaystyle
L_{0}f=Lf=\mathcal{L}f,\;\;f\in D(L_{0}),$ (4.107) $\displaystyle
L_{0}^{\prime}u=L_{0}u,\;\;u\in D(L_{0}^{\prime}).$

###### Proof.

The proof is straightforward and so will be omitted. ∎

######  Notation 4.21.

When we need to be explicit about the dependence of $L_{0}$ on $H$ and/or $J$
we will use the subscript $(\cdot)_{H}$ or $(\cdot)_{J,H}$ with these
operators, e.g., $(L_{0})_{H}$ and for the latter, we will just write
$L_{H,0}$ instead. Similarly, $(L_{0})_{J,H}$ will be written instead as
$L_{J,H,0}$.

The importance of the next two theorems becomes more clear by comparing it to
Theorem 3.11 with $A=L_{0},B=L_{H^{*}}$ as we use these results to prove
Theorem 4.24 below.

######  Theorem 4.22.

Let $I$ is a bounded interval. Then the following are true:
a) For $f\in D(L_{0})$ and $g\in D(L_{H^{*}})$ we have

(4.108) $\displaystyle\langle L_{0}f,g\rangle_{W}=\langle
f,L_{H^{*}}g\rangle_{W}.$

b) For $f\in D(L_{0})$ and $g\in D(L_{H^{*},0})$ we have

(4.109) $\displaystyle\langle L_{0}f,g\rangle_{W}=\langle
f,L_{H^{*},0}g\rangle_{W}.$

###### Proof.

The proof is similar to the proof of Theorem 4.9 and so it is omitted. ∎

######  Theorem 4.23.

Suppose $I$ is a bounded interval and the index-$1$ hypotheses (see Def. 4.14)
are true for $H,W$ with respect to $J$ on the interval $I$. Then

(4.110) $\displaystyle\operatorname{ran}L_{0}$ $\displaystyle=(\ker
L_{H^{*}})^{\perp},\;\;(\operatorname{ran}L_{0})^{\perp}=\ker L_{H^{*}},$
(4.111) $\displaystyle\ker L_{0}$
$\displaystyle=\\{0\\},\;\;\operatorname{ran}L_{H^{*}}=(\ker
L_{0})^{\perp}=L^{2}(I;W),$ (4.112) $\displaystyle\ker L_{H^{*}}$
$\displaystyle+\operatorname{ran}L_{0}=\operatorname{ran}L_{H^{*}}+\ker
L_{0}=L^{2}(I;W).$

In particular, $\operatorname{ran}L_{0}$ and $\ker L_{H^{*}}$ are closed
subspaces of $L^{2}(I;W)$, which are orthogonal to each other.

###### Proof.

Assume the hypotheses. Then by Corollary 4.16, the index-$1$ hypotheses are
also true for $H^{*},W$ with respect to $J$ on the interval $I$. First, we may
just assume that $I$ is a compact interval, for if it is not then we can just
take its closure instead which is now compact and the result for $I$ will then
follow from the statement for that compact interval. Second, (4.112) follows
immediately from (4.110) and (4.111). Third, it follows immediately from
Corollary 4.18 (applied to $L_{H^{*}}$) that
$\operatorname{ran}L_{H^{*}}=L^{2}(I;W)$ and by Corollary 4.18 (applied to
$L$) and Proposition 4.17.(ii) that $\ker L_{0}=\\{0\\}$, which proves the
equalities in (4.111). Next, the second identity of (4.110) follows from the
first identity since $\ker L_{H^{*}}$ is finite-dimensional by Corollary 4.18
(applied to $L_{H^{*}}$) and hence $(\ker
L_{H^{*}})^{{\perp}{\perp}}=\overline{\ker L_{H^{*}}}=\ker L_{H^{*}}$. Thus,
it remains to prove that $\operatorname{ran}L_{0}=(\ker L_{H^{*}})^{\perp}$
when $I$ is a compact interval, which we do so now.

Let $f\in L^{2}(I;W)$, where $I=[a,b]$ is a compact interval. Then, by
Proposition 4.17.(ii) and the hypotheses there is a unique solution $u\in
D(\mathcal{L})$ to the IVP $\mathcal{L}u=f,(Ju)(a)=0$, and, in addition, $u\in
L^{2}_{loc}(I;W)=L^{2}(I;W)$ so that $u\in D(L)$ with $Lu=\mathcal{L}u=f$. By
applying Corollary 4.18 to $L_{H^{*}}$ we know that
$\ker\mathcal{L}_{H^{*}}=\ker L_{H^{*}}$ and $\infty>\dim\ker
L_{H^{*}}=\operatorname{rank}J=:r>0$. Furthermore, using a similar proof as in
Corollary 4.18, we get a basis $\\{z_{i}:1\leq i\leq r\\}$ of $\ker L_{H^{*}}$
satisfying $(Jz_{i})(b)=JVe_{i},i=1,\ldots,r$, where $e_{1},\ldots,e_{n}$
denote the standard basis vectors for $\mathbb{C}^{n}$. Then it follows that

(4.113) $\displaystyle\langle f,z_{i}\rangle_{W}=\langle
Lu,z_{i}\rangle_{W}=\langle\mathcal{L}u,z_{i}\rangle_{W}-\langle
u,\mathcal{L}_{H^{*}}z_{i}\rangle_{W}$ (4.114)
$\displaystyle=\langle(Ju)(b),J^{+}(Jz_{i})(b)\rangle-\langle(Ju)(a),J^{+}(Jz_{i})(a)\rangle=\langle(Ju)(b),J^{+}(Jz_{i})(b)\rangle$
(4.115) $\displaystyle=\langle(Ju)(b),J^{+}JVe_{i}\rangle=\langle
JJ^{+}(Ju)(b),Ve_{i}\rangle=\langle(JJ^{+}Ju)(b),Ve_{i}\rangle$ (4.116)
$\displaystyle=\langle(Ju)(b),Ve_{i}\rangle=\langle
V^{*}(Ju)(b),e_{i}\rangle=\langle V^{-1}(Ju)(b),e_{i}\rangle$ (4.117)
$\displaystyle=\langle(V^{-1}JVV^{-1}u)(b),e_{i}\rangle=(V^{-1}JVV^{-1}u)(b)_{i}=(J_{11}u_{1})(b)_{i},$

for each $i=1,\ldots,r$, where $V=I_{n},J_{11}=J$ if $\det J\not=0$,
otherwise,

(4.118) $\displaystyle V^{-1}u=\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix},\;\;V^{-1}JVV^{-1}u=\begin{bmatrix}J_{11}u_{1}\\\
0\end{bmatrix}.$

Suppose that $f\in\operatorname{ran}L_{0}$. Then, for the above we must have
$u\in D(L_{0}),$ $L_{0}u=Lu=f,$ and $(Ju)(a)=(Ju)(b)=0$ so that by the above
calculation it follows that $\langle f,z_{i}\rangle=0$ for all $i=1,\ldots,r$.
As $z_{1},\ldots,z_{r}$ is a basis of solutions to $\ker L_{H^{*}}$ then this
implies that $\langle f,z\rangle_{W}=0$ for all $z\in\ker L_{H^{*}}$ and hence
$f\in(\ker L_{H^{*}})^{\perp}$. This proves that
$\operatorname{ran}L_{0}\subseteq(\ker L_{H^{*}})^{\perp}$. Conversely,
suppose $f\in(\ker L_{H^{*}})^{\perp}$. Then $\langle f,z_{i}\rangle_{W}=0$
for every $z\in\ker L_{H^{*}}$. By the above calculation for this $f$ and the
corresponding $u$, we have $0=\langle f,z_{i}\rangle=(J_{11}u_{1})(b)_{i}$ for
each $i=1,\ldots,r$ implying $(J_{11}u_{1})(b)=0$. As this implies $(Ju)(b)=0$
then it follows that $u\in D(L_{0})$ [as $u\in D(L)$ with $(Ju)(a)=(Ju)(b)=0$]
and so $L_{0}u=Lu=f$ which proves $f\in\operatorname{ran}L_{0}$. This proves
$(\ker L_{H^{*}})^{\perp}\subseteq\operatorname{ran}L_{0}$. Therefore,
$\operatorname{ran}L_{0}=(\ker L_{H^{*}})^{\perp}$, which completes the proof.
∎

######  Theorem 4.24.

Suppose $I$ is a bounded interval and the index-$1$ hypotheses (see Def. 4.14)
are true for $H,W$ with respect to $J$ on the interval $I$. Then the following
statements hold:
a) The subspaces $D(L_{0})$, $D(L_{H^{*},0})$, $D(L)$, and $D(L_{H^{*}})$ are
all dense in $L^{2}(I;W)$.
b) The operators $L_{0}:D(L_{0})\rightarrow L^{2}(I;W)$ and
$L_{H^{*}}:D(L_{H^{*}})\rightarrow L^{2}(I;W)$ are densely defined closed
operators with closed ranges and are adjoints of each other. In particular,

(4.119)
$\displaystyle(L_{0})^{*}=L_{H^{*}}=\overline{L_{H^{*}}}=(L_{H^{*}})^{**},\;\;L_{H^{*}}^{*}=L_{0}=\overline{L_{0}}=(L_{0})^{**}.$

c) The operators $L_{H^{*},0}:D(L_{H^{*},0})\rightarrow L^{2}(I;W)$ and
$L:D(L)\rightarrow L^{2}(I;W)$ are densely defined closed operators with
closed ranges and are adjoints of each other. In particular,

(4.120)
$\displaystyle(L_{H^{*},0})^{*}=L=\overline{L}=L^{**},\;\;L^{*}=L_{H^{*},0}=\overline{L_{H^{*},0}}=(L_{H^{*},0})^{**}.$

###### Proof.

Assume the hypotheses. Then by Corollary 4.16, the index-$1$ hypotheses are
also true for $H^{*},W$ with respect to $J$ on the interval $I$. First, by
duality (i.e., $H^{**}=H$) we need only prove the statements for $L_{0}$ and
$L_{H^{*}}$ as then the results for $L_{H^{*},0}$ and $L_{H}$ will follow
immediately by duality. Second, once we have proven that $L_{0}$ and
$L_{H^{*}}$ are densely defined closed operators then it follows immediately
from general results on adjoints [42, Theorem VIII.1.(b)] that the closure of
$L_{0}$ and $L_{H^{*}}$ (i.e., $\overline{L_{0}}$ and $\overline{L_{H^{*}}}$)
satisfies the relationships $L_{H^{*}}=\overline{L_{H^{*}}}=(L_{H^{*}})^{**}$
and $L_{0}=\overline{L_{0}}=(L_{0})^{**}$. Third, once we have proven that
$L_{0}$ is a densely defined it will follow immediately that $L$ is densely
defined [since $D(L_{0})\subseteq D(L)$] and then by duality (i.e.,
$H^{**}=H$) it follows that $L_{H^{*},0},L_{H^{*}}$ are also densely defined.
Fourth, if we can then prove that $(L_{0})^{*}=L_{H^{*}}$, it will follows
immediately from general results on adjoints [42, Theorem VIII.1] that
$L_{H^{*}}$ is a closed operator (so that $L_{H^{*}}=\overline{L_{H^{*}}})$,
$L_{0}$ is a closable operator, and $L_{H^{*}}^{*}=\overline{L_{0}}$. Finally,
if we can then prove $L_{0}$ is a closed operator, i.e.,
$\overline{L_{0}}=L_{0}$, we will have completed the proof of the theorem.

Thus, to prove the theorem we will now prove the following statements: (i)
$L_{0}$ is densely defined; (ii) $(L_{0})^{*}=L_{H^{*}}$; (iii) $L_{0}$ is a
closed operator. Again, as in the proof of Theorem 4.23, we can just assume in
the proof that $I$ is a compact interval as the results in the case of bounded
intervals will follow from this.

(i): Let $h\in D(L_{0})^{\perp}$. Then for any solution $g\in
D(\mathcal{L_{H^{*}}})$ of $\mathcal{L_{H^{*}}}g=h$ we have by Proposition
4.17.(ii) (applied to $H^{*}$ instead of $H$) that $g\in L^{2}_{loc}(I;W)$ and
since $I$ is compact we have $L^{2}_{loc}(I;W)=L^{2}(I;W)$. It now follows
that $g\in D(L_{H^{*}})$ and $L_{H^{*}}g=h$. Hence, from this and by Theorem
4.22.a), it follows that for every $f\in D(L_{0})$, we have

(4.121) $\displaystyle\langle L_{0}f,g\rangle_{W}=\langle
f,L_{H^{*}}g\rangle_{W}=\langle f,h\rangle_{W}=0.$

This implies that $g\in\operatorname{ran}(L_{0})^{\perp}$. By Theorem 4.23 we
know that $\ker L_{H_{*}}=(\operatorname{ran}L_{0})^{\perp}$ so that
$h=L_{H^{*}}g=0$. From which we conclude that $D(L_{0})^{\perp}=\\{0\\}$ and
thus
$\overline{D(L_{0})}=D(L_{0})^{{\perp}{\perp}}=\\{0\\}^{\perp}=L^{2}(I;W)$.
Therefore, $L_{0}$ is densely defined, which proves (i).

(ii): From part (i) we know that the linear operator
$L_{0}:D(L_{0})\rightarrow L^{2}(I;W)$ is densely defined and so now we
consider it’s adjoint $(L_{0})^{*}$. By Theorem 4.22.a), it follows that
$D(L_{H^{*}})\subseteq D((L_{0})^{*})$ and $(L_{0})^{*}g=L_{H^{*}}g$ for all
$g\in D(L_{H^{*}})$. Thus, to prove that $(L_{0})^{*}=L_{H^{*}}$, it suffices
to prove that $D((L_{0})^{*})\subseteq D(L_{H^{*}})$. Let $g\in
D((L_{0})^{*})$ and set $h=(L_{0})^{*}g$. Then, since we know by Theorem 4.23
that $\operatorname{ran}L_{H^{*}}=L^{2}(I;W)$, there exists an $f\in
D(L_{H^{*}})$ such that $L_{H^{*}}f=h$. Hence, from this and Theorem 4.22.a),
it follows that for every $u\in D(L_{0})$, we have

(4.122) $\displaystyle\langle L_{0}u,g\rangle_{W}=\langle
u,(L_{0})^{*}g\rangle_{W}=\langle u,L_{H^{*}}f\rangle_{W}=\langle
L_{0}u,f\rangle_{W}$

which implies $g-f\in(\operatorname{ran}L_{0})^{\perp}$. By Theorem 4.23 we
know that $(\operatorname{ran}L_{0})^{\perp}=\ker L_{H_{*}}\subseteq
D(L_{H_{*}})$ and hence $g-f\in D(L_{H_{*}})$. As $f\in D(L_{H^{*}})$ and
$D(L_{H^{*}})$ is a subspace of $L^{2}(I;W)$, it follows that $g\in
D(L_{H^{*}})$. Therefore, $D((L_{0})^{*})\subseteq D(L_{H^{*}})$, which proves
(ii).

(iii) From part (ii) we know that $D(L_{0})$ is dense in $L^{2}(I;W)$ and
since $D(L_{0})\subseteq D(L)$ then $D(L)$ is dense in $L^{2}(I;W)$. Hence, by
the hypotheses using $H^{*}$ instead of $H$, it follows that $D(L_{H^{*}})$ is
also dense in $L^{2}(I;W)$. Thus, as we proved $(L_{0})^{*}=L_{H^{*}}$, it
follows immediately from general results on adjoints [42, Theorem VIII.1] that
$L_{H^{*}}$ is a closed operator (so that $L_{H^{*}}=\overline{L_{H^{*}}})$,
$L_{0}$ is a closable operator, and $L_{H^{*}}^{*}=\overline{L_{0}}$. We will
now prove $\overline{L_{0}}=L_{0},$ i.e., $L_{0}$ is closed. Let
$\\{u_{m}\\}_{m\in\mathbb{N}}\subseteq D(L_{0})$ be a sequence converging in
$L^{2}(I;W)$ converging to $u$ such that $\\{L_{0}u_{m}\\}_{m\in\mathbb{N}}$
converges in $L^{2}(I;W)$ to $f$. If we can prove that $u\in D(L_{0})$ and
$L_{0}u=f$ then this will prove that $L_{0}$ is a closed operator. First,
since $L_{0}$ is closable and $L_{H^{*}}^{*}=\overline{L_{0}}$, it follows by
general results on closable operators [42, Proposition, p. 250] that $u\in
D(L_{H^{*}}^{*})$ and $L_{H^{*}}^{*}u=f$. Next, let $g\in\ker L_{H^{*}}$. Then
by Theorem 4.22.a) it follows that

(4.123) $\displaystyle\langle f,g\rangle_{W}=\lim_{m\rightarrow\infty}\langle
L_{0}u_{m},g\rangle_{W}=\lim_{m\rightarrow\infty}\langle
u_{m},L_{H^{*}}g\rangle_{W}=0$

which implies $f\in(\ker L_{H^{*}})^{\perp}=\operatorname{ran}L_{0}$, where
the latter equality follows from Theorem 4.23. It follows that there exists
$v\in D(L_{0})$ such that $L_{0}v=f=L_{H^{*}}^{*}u$. Hence, by Theorem 4.22.a)
it follows that for any $h\in D(L_{H^{*}})$, we have

(4.124) $\displaystyle\langle v,L_{H^{*}}h\rangle_{W}=\langle
L_{0}v,h\rangle_{W}=\langle L_{H^{*}}^{*}u,h\rangle_{W}=\langle
u,L_{H^{*}}h\rangle_{W}$

implying $u-v\in(\operatorname{ran}L_{H^{*}})^{\perp}=\ker L_{0}=\\{0\\}$,
where the latter two equalities follow from Theorem 4.23, and hence $u=v\in
D(L_{0})$ with $L_{0}u=L_{0}v=f$. Thus, we’ve proved $L_{0}$ is a closed
operator which proves (iii). Therefore, we have proven statements (i), (ii),
and (iii) which proves the theorem now from our discussion above. ∎

We now extend Theorem 4.24 to allow for arbitrary intervals
$I\subseteq\mathbb{R}$, which include unbounded intervals, e.g.,
$I=\mathbb{R}$, but under the local index-$1$ hypotheses now.

######  Remark 4.25.

The proof of the next theorem is essentially based on the notion of inductive
limits in the category of Hilbert spaces and unbounded operators. And,
although we do not elaborate more on this as it would require too much
additional background, we highly recommend the references [35, 36, 19], [21,
Exercises 11.5.26 and 11.5.27], [28, Appendix A], and [27] as a starting point
for the interested reader.

######  Theorem 4.26.

Suppose $I\subseteq\mathbb{R}$ is an interval (with nonempty interior) and the
local index-$1$ hypotheses (see Def. 4.14) are true for $H,W$ with respect to
$J$ on the interval $I$. Then the following statements hold:
i) The subspaces $D(L_{0}^{\prime}),D(L_{H^{*},0}^{\prime}),D(L),$ and
$D(L_{H^{*}})$ are all dense in $L^{2}(I;W)$.
ii) The operators $L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow L^{2}(I;W)$ and
$L_{H^{*}}:D(L_{H^{*}})\rightarrow L^{2}(I;W)$ are densely defined.
Furthermore, $L_{H^{*}}$ is a closed operator and $L_{0}$ is a closable
operator such that its closure, i.e.,
$\overline{L_{0}^{\prime}}=(L_{0}^{\prime})^{**}$, and $L_{H^{*}}$ are
adjoints of each other. In particular,

(4.125)
$\displaystyle(L_{0}^{\prime})^{*}=(\overline{L_{0}^{\prime}})^{*}=L_{H^{*}}=\overline{L_{H^{*}}}=(L_{H^{*}})^{**},\;\;(L_{H^{*}})^{*}=\overline{L_{0}^{\prime}}.$

iii) The operators $L_{H^{*},0}^{\prime}:D(L_{H^{*},0}^{\prime})\rightarrow
L^{2}(I;W)$ and $L:D(L)\rightarrow L^{2}(I;W)$ are densely defined.
Furthermore, $L$ is a closed operator and $L_{H^{*},0}^{\prime}$ is a closable
operator such that its closure, i.e.,
$\overline{L_{H^{*},0}^{\prime}}=(L_{H^{*},0}^{\prime})^{**}$, and $L$ are
adjoints of each other. In particular,

(4.126)
$\displaystyle(L_{H^{*},0}^{\prime})^{*}=(\overline{L_{H^{*},0}^{\prime}})^{*}=L=\overline{L}=L^{**},\;\;L^{*}=\overline{L_{H^{*},0}^{\prime}}.$

###### Proof.

First, as the hypotheses are valid for both $H$ and $H^{*}$ (by Corollary
4.16), and $(H^{*})^{*}=H$, then we need only prove the statement for
$L_{0}^{\prime}$ and $L_{H^{*}}$ as the other results will follow immediately
by this duality. Second, once we have proven that $L_{0}^{\prime}$ and
$L_{H^{*}}$ are densely defined operators with $L_{0}^{\prime}$ and
$L_{H^{*}}$ a closable and closed operator, respectively, then it follows
immediately from general results that the closure of $L_{0}^{\prime}$ and
$L_{H^{*}}$ satisfy the relationships
$L_{H^{*}}=\overline{L_{H^{*}}}=(L_{H^{*}})^{**}$ and
$\overline{L_{0}^{\prime}}=(L_{0}^{\prime})^{**}$ and hence once we prove
$(L_{0}^{\prime})^{*}=L_{H^{*}}$, then it will follow immediately that
$(L_{H^{*}})^{*}=\overline{L_{0}^{\prime}}$. On the other hand, once we know
that $D(L_{0}^{\prime})$ is dense for any arbitrary $H$ satisfying the local
index-1 hypotheses then by duality $D(L_{H^{*},0}^{\prime})$ is also dense so
that since $D(L_{H^{*},0}^{\prime})\subseteq D(L_{H^{*}})$, it will follow
that $D(L_{H^{*}})$ is dense which implies then by Theorem 4.9.a) that the
adjoint $(L_{0}^{\prime})^{*}$ of $L_{0}^{\prime}$ is an extension of
$L_{H^{*}}$, i.e., $D(L_{H^{*}})\subseteq D((L_{0}^{\prime})^{*})$ with
$L_{H^{*}}f=(L_{0}^{\prime})^{*}f$ for every $f\in D(L_{H^{*}})$ [which we
denote by $L_{H^{*}}\subset(L_{0}^{\prime})^{*}$], so that
$(L_{0}^{\prime})^{*}$ is densely defined from which it follows from general
results on adjoints that $L_{0}^{\prime}$ is closable, and finally since
$L_{H^{*}}\subset(L_{0}^{\prime})^{*}$ then to prove
$(L_{0}^{\prime})^{*}=L_{H^{*}}$ (which implies from this by general results
that $L_{H^{*}}$ is closed), we need only prove $(L_{0}^{\prime})^{*}\subset
L_{H^{*}}$. Thus, to recap, to complete the proof we need only prove that
$D(L_{0}^{\prime})$ is dense for any arbitrary $H$ satisfying the local
index-1 hypotheses and that $(L_{0}^{\prime})^{*}\subset L_{H^{*}}$, i.e.,
$D((L_{0}^{\prime})^{*})\subseteq D(L_{H^{*}})$ and
$L_{H^{*}}f=(L_{0}^{\prime})^{*}f$ for every $f\in D((L_{0}^{\prime})^{*})$.
We will do this next.

For any arbitrary $\alpha,\beta\in\mathbb{R}$ with $\alpha<\beta$ such that
$\triangle=[\alpha,\beta]\subseteq\operatorname{int}I$, we consider
$\mathcal{L}$ on the compact interval $\triangle$. Denote by
$L_{\triangle,0},L_{\triangle}$ the closed minimal and maximal operators
generated by $\mathcal{L}$ in $L^{2}(\triangle;W)$. With abuse of notation, we
can treat any element of $D(L_{\triangle,0})$ as an element of
$D(L_{0}^{\prime})$ by zero extension, i.e., setting it equal to zero on
$I\setminus\triangle$. It then follows that

(4.127) $\displaystyle\cup_{\triangle\subseteq
I}D(L_{\triangle,0})=D(L_{0}^{\prime}).$

As each $D(L_{\triangle,0})$ is dense in $L^{2}(\triangle;W)$ it then follows
that $D(L_{0}^{\prime})$ is dense in $L^{2}(I;W)$. Hence as
$D(L_{0}^{\prime})\subseteq D(L)$ with $L_{0}f=Lf$ for all $f\in
D(L_{0}^{\prime})$, it follows that both the linear operators
$L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow L^{2}(I;W)$ and
$L:D(L)\rightarrow L^{2}(I;W)$ are densely defined with $L_{0}^{\prime}\subset
L$. It now follows by duality, using the same argument but with $H^{*}$
instead of $H$, that both the linear operators
$L_{H^{*},0}^{\prime}:D(L_{H^{*},0}^{\prime})\rightarrow L^{2}(I;W)$ and
$L:D(L_{H^{*}})\rightarrow L^{2}(I;W)$ are densely defined. Hence, as
mentioned above, to complete the proof we need only prove that
$(L_{0}^{\prime})^{*}\subset L_{H^{*}}$. We now write
$\langle\cdot,\cdot\rangle_{\triangle}$ for the inner product in
$L^{2}(\triangle;W)$; by $h_{\triangle}$ we denote the restriction of a
function $h$ from $L^{2}(I;W)$ to $L^{2}(\triangle;W)$ (in the natural way).
We already know that $L_{\triangle,0}^{*}=L_{H^{*},\triangle}$ and
$L_{\triangle,0}\subset L_{0}^{\prime}$ (in the abuse of notation sense
above), hence $L_{\triangle,0}\subset L_{0}^{\prime}\subset L$. We also know
that for every $h\in D(L)$ we have $h_{\triangle}\in D(L_{\triangle})$. Let
$f\in D((L_{0}^{\prime})^{*})$. Then for any $g\in L^{2}(I;W)$ with $g\in
D(L_{\triangle,0})$ [so that $g\in D(L_{0}^{\prime})$ in the sense above] it
follows that

(4.128)
$\displaystyle\langle((L_{0}^{\prime})^{*}f)_{\triangle},g_{\triangle}\rangle_{\triangle}=\langle(L_{0}^{\prime})^{*}f,g\rangle=\langle
f,L_{0}^{\prime}g\rangle=\langle f,L_{\triangle,0}g\rangle=\langle
f_{\triangle},L_{\triangle,0}g_{\triangle}\rangle_{\triangle}.$

This implies that $f_{\triangle}\in D(L_{\triangle,0}^{*})$ and, as
$L_{\triangle,0}^{*}=L_{H^{*},\triangle}$, this also implies

(4.129)
$\displaystyle((L_{0}^{\prime})^{*}f)_{\triangle}=L_{\triangle,0}^{*}f_{\triangle}=L_{H^{*},\triangle}f_{\triangle}=(L_{H^{*}}f)_{\triangle}.$

As this is true for any compact interval $\triangle$ with nonempty interior
satisfying $\triangle\subseteq\operatorname{int}I$ then this implies that
$f\in D(\mathcal{L}_{H^{*}})$ and

(4.130) $\displaystyle\mathcal{L}_{H^{*}}f=(L_{0}^{\prime})^{*}f\in
L^{2}(I;W),$

which implies $f\in D(L_{H^{*}})$ and
$L_{H^{*}}f=\mathcal{L}_{H^{*}}f=(L_{0}^{\prime})^{*}f$. This proves that
$(L_{0}^{\prime})^{*}\subset L_{H^{*}}$, which completes the proof of the
theorem. ∎

If the hypotheses of Theorem 4.26 are true then this theorem tells us
$L_{0}^{\prime}$ is a densely defined closable operator. As such, we know that
its closure $\overline{L_{0}^{\prime}}=(L_{0}^{\prime})^{**}$ is the smallest
closed operator extension of $L_{0}^{\prime}$ (see [42, Sec. VIII.2]). Because
of this we can extend the Definition 4.19 to unbounded intervals as follows.

######  Definition 4.27.

In the case $I$ is an unbounded interval and the local index-$1$ hypotheses
(see Def. 4.14) are true for $H,W$ with respect to $J$ on the interval $I$,
the closed minimal operator $L_{0}$ generated by $\mathcal{L}$ is defined by

(4.131) $\displaystyle
L_{0}:=\overline{L_{0}^{\prime}}=(L_{0}^{\prime})^{**}.$

The next theorem shows in what sense this definition is an extension of the
definition of $L_{0}$ in Definition 4.19 from bounded intervals to unbounded
intervals.

######  Theorem 4.28.

If $I$ is a bounded interval and the index-$1$ hypotheses (see Def. 4.14) are
true for $H,W$ with respect to $J$ on the interval $I$, then

(4.132) $\displaystyle L_{0}=\overline{L_{0}^{\prime}}=(L_{0}^{\prime})^{**}.$

###### Proof.

Assume the hypotheses. Then the local index-$1$ hypotheses (see Def. 4.14) are
also true for $H^{*},W$ with respect to $J$ on the interval $I$ (by Corollary
4.16). Thus, by Theorem 4.26 we know that
$L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow L^{2}(I;W)$ is a densely defined
closable operator and hence it follows that
$\overline{L_{0}^{\prime}}=(L_{0}^{\prime})^{**}$ (see [42, Sec. VIII.2]).
Now, by Theorem 4.24 we have $L_{H^{*}}^{*}=L_{0}$ and by Theorem 4.26 we know
that $(L_{0}^{\prime})^{*}=L_{H^{*}}$. Therefore,
$L_{0}=L_{H^{*}}^{*}=(L_{0}^{\prime})^{**}=\overline{L_{0}^{\prime}}$. ∎

One of the main problems in the spectral theory of DAEs is to answer the
following question: If $H^{*}=H$, what additional hypotheses imply the maximal
operator $L:D(L)\rightarrow L^{2}(I;W)$ generated by $\mathcal{L}$ is a self-
adjoint operator, i.e., $L^{*}=L$? The goal of the next section is to consider
this question under the additional assumption that $H$ is periodic.

##  5\. Spectral theory for linear differential-algebraic equations with
periodic coefficients

In this section we will study the linear differential-algebraic equations with
periodic coefficients [say, $d$-periodic for some fixed period
$d\in(0,\infty)$], i.e., $d$-periodic linear DAEs (4.1), in terms of the
spectral problem (1.1) and the corresponding spectral theory of their
associated DA operators $\mathcal{L},L_{0}^{\prime},L_{0},L$ on the Hilbert
space $L^{2}(\mathbb{R};W)$. Thus, we continue to assume that (4.3)-(4.5) are
true for $J,H,W$ with the interval $I=(-\infty,\infty)=\mathbb{R}$, but now we
have the addition periodicity hypothesis:

(5.1) $\displaystyle H(t+d)=H(t),\;W(t+d)=W(t),\;\text{for all
}t\in\mathbb{R}.$

######  Remark 5.1.

It should be remarked that our conclusions below are unaffected by weakening
the latter hypothesis from “for all $t\in\mathbb{R}$” to “for a.e.
$t\in\mathbb{R}$.”

In order to proceed in the more general fashion for the self-adjoint spectral
theory, we eventually need to make use of the following additional hypotheses
on $H$ and on the minimal and maximal operators
$L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow L^{2}(\mathbb{R};W)$ and
$L:D(L)\rightarrow L^{2}(\mathbb{R};W)$, respectively, generated by the DA
operator $\mathcal{L}:D(\mathcal{L})\rightarrow[\mathcal{M}(\mathbb{R})]^{n}$
[see (4.6)-(4.8) and Def. 4.5]:

(5.2) $\displaystyle H(t)^{*}=H(t),\;\text{for a.e.\ }t\in(0,d).$ (5.3)
$D(L_{0}^{\prime})$ is dense in $L^{2}(\mathbb{R};W)$. (5.4)
$\displaystyle(L_{0}^{\prime})^{*}=L.$

######  Remark 5.2.

There are some remarks in order regarding the hypotheses (5.3), (5.4), and how
they will be used to prove $L$ is self-adjoint under the hypothesis (5.2).
First, by Lemma 4.7 we know that $D(L_{0}^{\prime}),D(L)$ are subspaces of
$L^{2}(\mathbb{R};W)$ with $D(L_{0}^{\prime})\subseteq D(L)$ and
$L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow L^{2}(\mathbb{R};W)$ and
$L:D(L)\rightarrow L^{2}(\mathbb{R};W)$ defined by $Lg=\mathcal{L}g$ for $g\in
D(L)$ and $L_{0}^{\prime}f=\mathcal{L}f=Lf$ for $f\in D(L_{0}^{\prime})$ are
well-defined linear operators. Second, if (5.3) is true, i.e.,
$L_{0}^{\prime}$ is densely defined, then $L$ is also densely defined since
$L_{0}^{\prime}\subset L$, and also by Theorem 4.9 we know $L_{0}^{\prime}$ is
symmetric, hence closable, with
$L\subset(L_{0}^{\prime})^{*}=(\overline{L_{0}^{\prime}})^{*}$ which implies
from this and the fact $(L_{0}^{\prime})^{*}$ is closed that $L$ must also be
closable with
$L\subset\overline{L}\subset(L_{0}^{\prime})^{*}=(\overline{L_{0}^{\prime}})^{*}$.
Third, it follows from this that if (5.3) is true then (5.4) is equivalent to
$D((L_{0}^{\prime})^{*})\subseteq D(L)$, in which case
$(\overline{L_{0}^{\prime}})^{*}=(L_{0}^{\prime})^{*}=L$. Fourth, if (5.3) is
true then by the corollary to [42, Theorem VIII.3] and by [41, Theorem X.1],
the following two conditions are equivalent: a) $L_{0}^{\prime}$ is
esssentially self-adjoint [i.e., it’s closure
$\overline{L_{0}^{\prime}}=(L_{0}^{\prime})^{**}$ is self-adjoint]; b)
$\ker((L_{0}^{\prime})^{*}-zI)=\ker((L_{0}^{\prime})^{*}-\overline{z}I)=\\{0\\}$
for some $z\in\mathbb{C}\setminus\mathbb{R}$. Hence, if (5.3) and (5.4) are
true then the following two conditions are equivalent: a) $L$ is self-adjoint;
b) $\ker(L-zI)=\ker(L-\overline{z}I)=\\{0\\}$ for some
$z\in\mathbb{C}\setminus\mathbb{R}$. Therefore, the goal of this section is to
find additional “natural” hypotheses besides (5.2), (5.3), and (5.4) that
allow us to conclude that $\ker(L-zI)=\ker(L-\overline{z}I)=\\{0\\}$ for some
$z\in\mathbb{C}\setminus\mathbb{R}$ as then from this remark we will have
proven that $L$ is self-adjoint.

######  Definition 5.3.

For any $m,n\in\mathbb{N}$, we will need in our analysis below, the left ($+$)
and right ($-$) translation (or shift) operators
$U_{\pm}:M_{m,n}(\mathcal{M}(\mathbb{R}))\rightarrow
M_{m,n}(\mathcal{M}(\mathbb{R}))$, which are the linear operators that are
inverses of each other and defined by

$\displaystyle f\in M_{m,n}(\mathcal{M}(\mathbb{R})),\;(U_{\pm}f)(t)=f(t\pm
d),\;\forall t\in\mathbb{R},$

with equality of functions in the usual sense of equal a.e. on $\mathbb{R}$.

The reader can verify that these translation operators are well-defined
functions which are linear and inverses of each other. Additional properties
of these translation operators, that are fundamental in our studies, are given
in the following lemma and proposition which are stated in terms of the DA
operator $\mathcal{L}$ and the minimal and maximal operators $L_{0}^{\prime}$
and $L$, respectively.

######  Lemma 5.4.

The subspace $D(\mathcal{L})$ of $[\mathcal{M}(\mathbb{R})]^{n}$ is an
invariant subspace of the translation operators
$U_{\pm}:[\mathcal{M}(\mathbb{R})]^{n}\rightarrow[\mathcal{M}(\mathbb{R})]^{n}$
and

(5.5) $\displaystyle U_{\pm}(D(\mathcal{L}))=D(\mathcal{L}).$

Moreover, these translation operators commute with the DA operator
$\mathcal{L}:D(\mathcal{L})\rightarrow[\mathcal{M}(\mathbb{R})]^{n}$, i.e.,

(5.6) $\displaystyle\mathcal{L}U_{\pm}=U_{\pm}\mathcal{L}.$

In addition, $\ker\mathcal{L}$ is an invariant subspace of these translation
operators and

(5.7) $\displaystyle U_{\pm}(\ker\mathcal{L})=\ker\mathcal{L}.$

###### Proof.

First, note that $D(\mathcal{L})$ is a subspace of
$[\mathcal{M}(\mathbb{R})]^{n}$ and
$U_{\pm}:[\mathcal{M}(\mathbb{R})]^{n}\rightarrow[\mathcal{M}(\mathbb{R})]^{n}$
are invertible linear operators on $[\mathcal{M}(\mathbb{R})]^{n}$ which are
inverses of each other. Thus, if we prove that
$U_{\pm}(D(\mathcal{L}))\subseteq D(\mathcal{L})$ then it will follow
immediately that $D(\mathcal{L})$ is an invariant subspace of $U_{\pm}$ and
$U_{\pm}(D(\mathcal{L}))=D(\mathcal{L})$. We will now prove that
$U_{\pm}(D(\mathcal{L}))\subseteq D(\mathcal{L})$. Let $f\in D(\mathcal{L})$
or, equivalently, $f\in[\mathcal{M}(\mathbb{R})]^{n}$ and
$Jf\in[W^{1,1}_{loc}(\mathbb{R})]^{n}$. Then it follows immediately that
$U_{\pm}f\in[\mathcal{M}(\mathbb{R})]^{n}$ and
$J(U_{\pm}f)=U_{\pm}(Jf)\in[W^{1,1}_{loc}(\mathbb{R})]^{n}$ since
$U_{\pm}:[W^{1,1}_{loc}(\mathbb{R})]^{n}\rightarrow[W^{1,1}_{loc}(\mathbb{R})]^{n}$,
that is, the subspace $[W^{1,1}_{loc}(\mathbb{R})]^{n}$ of
$[\mathcal{M}(\mathbb{R})]^{n}$ is an invariant subspace of
$U_{\pm}:[\mathcal{M}(\mathbb{R})]^{n}\rightarrow[\mathcal{M}(\mathbb{R})]^{n}$.
This implies that $U_{\pm}f\in D(\mathcal{L})$. Thus we have proven
$U_{\pm}(D(\mathcal{L}))\subseteq D(\mathcal{L})$, as desired. Next, we prove
that

(5.8) $\displaystyle\mathcal{L}(U_{\pm}f)=U_{\pm}(\mathcal{L}f),\;\;\forall
f\in D(\mathcal{L}).$

Let $f\in D(\mathcal{L})$. Then $\mathcal{L}f\in[\mathcal{M}(\mathbb{R})]^{n}$
and, as we have just proved, $U_{\pm}f\in D(\mathcal{L})$ so that
$\mathcal{L}(U_{\pm}f),U_{\pm}(\mathcal{L}f)\in[\mathcal{M}(\mathbb{R})]^{n}$.
We will now prove that $\mathcal{L}(U_{\pm}f)$ and $U_{\pm}(\mathcal{L}f)$ are
equal as elements in $[\mathcal{M}(\mathbb{R})]^{n}$. First, it follows from
the assumption (5.1) that

(5.9) $\displaystyle
U_{\pm}(Hf)=(U_{\pm}H)(U_{\pm}f)=H(U_{\pm}f),\;U_{\pm}(Wf)=(U_{\pm}W)(U_{\pm}f)=W(U_{\pm}f)$

in $[\mathcal{M}(\mathbb{R})]^{n}$. It also then follows by hypotheses (4.5)
that

(5.10) $\displaystyle
U_{\pm}(W^{-1}f)=(U_{\pm}W^{-1})(U_{\pm}f)=W^{-1}(U_{\pm}f)$

in $[\mathcal{M}(\mathbb{R})]^{n}$. Hence, it follows that

(5.11) $\displaystyle
U_{\pm}(\mathcal{L}f)=U_{\pm}W^{-1}\left(\frac{d}{dt}Jf+Hf\right)=W^{-1}\left(\frac{d}{dt}J(U_{\pm}f)+H(U_{\pm}f)\right)=\mathcal{L}(U_{\pm}f)$

in $[\mathcal{M}(\mathbb{R})]^{n}$. Thus, to complete the proof it remains
only to prove that $U_{\pm}(\ker\mathcal{L})=\ker\mathcal{L}$. Again as
$U_{+}$ and $U_{-}$ are inverses of each other, we need only prove that
$U_{\pm}(\ker\mathcal{L})\subseteq\ker\mathcal{L}$. Let $f\in\ker\mathcal{L}$.
Then it follows from the above that

(5.12)
$\displaystyle\mathcal{L}(U_{\pm}f)=(\mathcal{L}U_{\pm})f=(U_{\pm}\mathcal{L})f=U_{\pm}(\mathcal{L}f)=U_{\pm}0=0$

and so $U_{\pm}f\in\ker\mathcal{L}$. This proves that
$U_{\pm}(\ker\mathcal{L})\subseteq\ker\mathcal{L}$, which completes the proof.
∎

######  Proposition 5.5.

The subspaces $D(L_{0}^{\prime})$ and $D(L)$ of $L^{2}(\mathbb{R};W)$ are
invariant subspaces of the translation operators
$U_{\pm}:L^{2}(\mathbb{R};W)\rightarrow L^{2}(\mathbb{R};W)$ and

(5.13) $\displaystyle
U_{\pm}(D(L_{0}^{\prime}))=D(L_{0}^{\prime}),\;\;U_{\pm}(D(L))=D(L).$

Moreover, these translation operators commute with the minimal and maximal
operators $L_{0}^{\prime}:D(L_{0}^{\prime})\to L^{2}(\mathbb{R};W)$ and
$L:D(L)\to L^{2}(\mathbb{R};W)$, respectively, i.e.,

(5.14) $\displaystyle
L_{0}^{\prime}U_{\pm}=U_{\pm}L_{0}^{\prime},\;\;LU_{\pm}=U_{\pm}L.$

In addition, $\ker L_{0}^{\prime}$ and $\ker L$ are invariant subspace of
these translation operators and

(5.15) $\displaystyle U_{\pm}(\ker L_{0}^{\prime})=\ker
L_{0}^{\prime},\;\;U_{\pm}(\ker L)=\ker L.$

###### Proof.

First, it follows immediately from the hypotheses (4.5) and (5.1) that the
subspace $L^{2}(\mathbb{R};W)$ of $[\mathcal{M}(\mathbb{R})]^{n}$ is an
invariant subspace of the translation operators
$U_{\pm}:[\mathcal{M}(\mathbb{R})]^{n}\rightarrow[\mathcal{M}(\mathbb{R})]^{n}$
and their restriction $U_{\pm}:L^{2}(\mathbb{R};W)\rightarrow
L^{2}(\mathbb{R};W)$ are linear operators which are inverses of each other.
Second, on $D(L_{0}^{\prime})$ we know that $L_{0}^{\prime}=\mathcal{L}$ and
on $D(L)$ we know that $L=\mathcal{L}$. It follows from this and Lemma 5.4
that to prove the proposition we need only prove that

(5.16) $\displaystyle U_{\pm}(D(L_{0}^{\prime}))\subseteq
D(L_{0}^{\prime}),\;\;U_{\pm}(D(L))\subseteq D(L).$

Let $f\in D(L)$. Then $f\in L^{2}(\mathbb{R};W),$ $f\in D(\mathcal{L}),$ and
$\mathcal{L}f\in L^{2}(\mathbb{R};W)$. It follows from this and Lemma 5.4 that

(5.17) $\displaystyle U_{\pm}f\in L^{2}(\mathbb{R};W),\;\;U_{\pm}f\in
D(\mathcal{L}),\;\;\mathcal{L}(U_{\pm}f)=U_{\pm}(\mathcal{L}f)\in
L^{2}(\mathbb{R};W)$

which implies $U_{\pm}f\in D(L)$. This proves $U_{\pm}(D(L))\subseteq D(L)$.
Similarly, let $f\in D(L_{0}^{\prime})$. Then $f\in D(L)$ and $Jf$ has compact
support in $\mathbb{R}$. It follows that $U_{\pm}f\in D(L)$ and
$J(U_{\pm}f)=U_{\pm}(Jf)$ has compact support in $\mathbb{R}$ which implies
$U_{\pm}f\in D(L_{0}^{\prime})$. This proves
$U_{\pm}(D(L_{0}^{\prime}))\subseteq D(L_{0}^{\prime})$. Therefore, the
proposition is proved. ∎

The next theorem is our main result on the spectral theory of
$\mathcal{L},L_{0}^{\prime},$ and $L$ in regards to eigenvalues of finite
multiplicities, in particular, whether they can have them or not. As it turns
out, $L_{0}^{\prime}$ and $L$ do not, whereas $\mathcal{L}$ can and in which
case additional important information can be extracted. The key to the
analysis is the commutivity of these operators with the translation operators
$U_{\pm}$.

######  Theorem 5.6.

Let $z\in\mathbb{C}$. Then following statements are true:
(a)
$\\{0\\}\subseteq\ker(L_{0}^{\prime}-zI)\subseteq\ker(L-zI)\subseteq\ker(\mathcal{L}-zI)$.
(b) If $\dim\ker(\mathcal{L}-zI)<\infty$ then either
$\ker(\mathcal{L}-zI)=\\{0\\}$, in which case
$\\{0\\}=\ker(L_{0}^{\prime}-zI)=\ker(L-zI)=\ker(\mathcal{L}-zI)$, or there
exists $f\in\ker(\mathcal{L}-zI)$ which is an eigenvector of the left
translation operator
$U_{+}:[\mathcal{M}(\mathbb{R})]^{n}\rightarrow[\mathcal{M}(\mathbb{R})]^{n}$
with a nonzero eigenvalue $\lambda$, i.e.,

(5.18) $\displaystyle\mathcal{L}f=zf,\;\;U_{+}f=\lambda
f,\;\;f\not=0,\;\;\lambda\not=0,$

for some $\lambda\in\mathbb{C}$.
(c) The translation operators $U_{\pm}:L^{2}(\mathbb{R};W)\rightarrow
L^{2}(\mathbb{R};W)$ are unitary operators which are inverses of each other
and have no eigenvalues.
(d) If $\dim\ker(L-zI)<\infty$ then
$\ker(L_{0}^{\prime}-zI)=\ker(L-zI)=\\{0\\}$.
(e) If $\dim\ker(L_{0}^{\prime}-zI)<\infty$ then
$\ker(L_{0}^{\prime}-zI)=\\{0\\}$.

###### Proof.

Let $z\in\mathbb{C}$. As the proof for $z\not=0$ is the same for the proof
with $z=0$ (as we can just replace $H$ with $H-zW$ and use the fact that
$\mathcal{L}_{H}-zI=\mathcal{L}_{H-zW},L_{H}-zI=L_{H-zW},L_{H,0}^{\prime}-zI=L_{H-zW,0}^{\prime}$),
then we can assume without loss of generality that $z=0$.

(a): This follows immediately from the fact that
$\mathcal{L},L,L_{0}^{\prime}$ are linear operators on their respective
domains $D(\mathcal{L}),D(L),D(L_{0}^{\prime})$ and by their definitions we
have that $Lf=\mathcal{L}f$ for every $f\in D(L)$ and
$L_{0}^{\prime}f=\mathcal{L}f$ for every $f\in D(L_{0}^{\prime})$ with
$D(L_{0}^{\prime})\subseteq D(L)\subseteq D(\mathcal{L})$.

(b): By Lemma 5.4 we know that $U_{\pm}:\ker\mathcal{L}\to\ker\mathcal{L}$ are
well-defined linear operators on the vector space $\ker L$ over the field
$\mathbb{C}$ which are inverses of each other. Hence, if
$\dim\ker\mathcal{L}<\infty$ and $\ker\mathcal{L}\not=\\{0\\}$ then this
implies by elementary linear algebra that there exists an eigenvector
$f\in\ker\mathcal{L}$ and corresponding eigenvalue $\lambda\in\mathbb{C}$ of
$U_{+}$ and since $U_{+}$ is invertible then $\lambda\not=0$. This proves
either $\ker\mathcal{L}=\\{0\\}$, in which case it follows from (a) that
$\\{0\\}=\ker L_{0}^{\prime}=\ker L=\ker\mathcal{L}$, or there exists
$f\in\ker L$ which is an eigenvector of the left translation operator $U_{+}$
with a nonzero eigenvalue $\lambda$.

(c): First, its obvious that $U_{\pm}:L^{2}(\mathbb{R};W)\to
L^{2}(\mathbb{R};W)$ are well-defined linear operators which are inverses of
each other. Thus, to prove $U_{-}$ is unitary it suffices to prove that its
inverse $U_{+}$ is unitary. Let $f,g\in L^{2}(\mathbb{R};W)$. Then

$\displaystyle\langle U_{+}f,U_{+}g\rangle_{W}=\int_{\mathbb{R}}\langle
W(t)(U_{+}f)(t),(U_{+}g)(t)\rangle dt=\int_{\mathbb{R}}\langle
W(t)f(t+d),g(t+d)\rangle dt$ $\displaystyle=\int_{\mathbb{R}}\langle
W(t-d)f(t),g(t)\rangle dt=\int_{\mathbb{R}}\langle W(t)f(t),g(t)\rangle
dt=\langle U_{+}f,U_{+}g\rangle_{W}.$

This proves $U_{-}$ is a unitary operator on $L^{2}(\mathbb{R};W)$. Next, as
$U_{-}$ is the inverse of the unitary operator $U_{+}$ then to show the former
has no eigenvalues it suffices to show the latter has no eigenvalues. Also,
$U_{+}$ is a unitary operator so if it did have an eigenvalue
$\lambda\in\mathbb{C}$ then it follows that $|\lambda|=1$. Hence, suppose
$f\in L^{2}(\mathbb{R};W)$ and $\lambda\in\mathbb{C}$ with $|\lambda|=1$ such
that $U_{+}f=\lambda f.$ We will prove that $f=0$, which will complete the
proof of statement (c). As the equalities
$\lambda^{m}f(\cdot)=(U_{+}^{m}f)(\cdot)=f(\cdot+md)$ hold as elements in
$L^{2}(\mathbb{R};W)$ for any $m\in\mathbb{Z}$ then

$\displaystyle\infty>\langle f,f\rangle_{W}=\int_{\mathbb{R}}\langle
W(t)f(t),f(t)\rangle dt=\sum_{m\in\mathbb{Z}}\int_{[md,(m+1)d]}\langle
W(t)f(t),f(t)\rangle dt$
$\displaystyle=\sum_{m\in\mathbb{Z}}\int_{[0,d]}\langle
W(t+md)f(t+md),f(t+md)\rangle
dt=\sum_{m\in\mathbb{Z}}\int_{[0,d]}\langle\lambda^{m}W(t)f(t),\lambda^{m}f(t)\rangle
dt$ $\displaystyle=\sum_{m\in\mathbb{Z}}\int_{[0,d]}\langle
W(t)f(t),f(t)\rangle dt$

which implies $\int_{[0,d]}\langle W(t)f(t),f(t)\rangle dt=0$ and hence
$\langle f,f\rangle_{W}=0$ so that $f=0$, as desired.

(d): By Proposition 5.5 we know that $U_{+}:\ker L\to\ker L$ is a linear
operator on the vector space $\ker L$ over the field $\mathbb{C}$, which
commutes with the operator $L$, and is the restriction of the operator
$U_{+}:L^{2}(\mathbb{R};W)\rightarrow L^{2}(\mathbb{R};W)$ to the invariant
subspace $\ker L$. Hence, if $\dim\ker L<\infty$ and $\ker L\not=\\{0\\}$ then
this implies by elementary linear algebra of the existence of an eigenvalue of
$U_{+}:\ker L\to\ker L$ and hence of $U_{+}:L^{2}(\mathbb{R};W)\rightarrow
L^{2}(\mathbb{R};W)$, a contradiction of (c). This proves that if $\dim\ker
L<\infty$ then $\ker L=\\{0\\}$ and hence by (a), $\ker L_{0}^{\prime}=\ker
L=\\{0\\}$.

(e): The proof of (e) follows from Proposition 5.5 using a similar argument as
the proof of (d). ∎

######  Remark 5.7.

Any function $f$ and scalar $\lambda=e^{ikd}$ satisfying the eigenvalue
problem (5.18) with $z\in\mathbb{C}$ is called a Bloch solution and Floquet
multiplier, respectively, of the periodic DAEs $\mathcal{L}$ in which case the
scalar $k$ is called a wavenumber. The multivalued function $z=z(k)$ of $k$ is
then referred to as the (complex) dispersion relation.

The next theorem is key to proving our main result in this paper on the self-
adjoint spectral theory of periodic DAEs.

######  Theorem 5.8.

Suppose (5.2), (5.3), and (5.4) are true. If there exists
$z_{0}\in\mathbb{C}\setminus\mathbb{R}$ such that

(5.19) $\displaystyle\dim\ker(\mathcal{L}-z_{0}I)<\infty\text{ and
}\dim\ker(\mathcal{L}-\overline{z_{0}}I)<\infty$

or

(5.20) $\displaystyle\dim\ker(L-z_{0}I)<\infty\text{ and
}\dim\ker(L-\overline{z_{0}}I)<\infty$

then

(5.21) $\displaystyle\ker(L-z_{0}I)=\ker(L-\overline{z_{0}}I)=\\{0\\},$ (5.22)
$\displaystyle L:D(L)\to L^{2}(\mathbb{R};W)\text{ is a self-adjoint operator
on }L^{2}(\mathbb{R};W),$

and

(5.23) $\displaystyle L\text{ has no eigenvalues of finite multiplicity.}$

###### Proof.

First, by Theorem 5.6.(d) we know that $L$ has no eigenvalues of finite
multiplicity. Second, by Theorem 5.6.(a) it suffices to prove the statement
assuming $\dim\ker(L-z_{0}I)<\infty$ and
$\dim\ker(L-\overline{z_{0}}I)<\infty$ for some
$z_{0}\in\mathbb{C}\setminus\mathbb{R}$. Thus, assume this is true. Then
$\ker(L-z_{0}I)=\ker(L-\overline{z_{0}}I)=\\{0\\}$ since $L$ has no
eigenvalues of finite multiplicity. Assume also that (5.2), (5.3), and (5.4)
are true. It now follows from this by Remark 5.2 that $L:D(L)\to
L^{2}(\mathbb{R};W)$ is a self-adjoint operator on $L^{2}(\mathbb{R};W)$. This
proves the theorem. ∎

The following is the main result of this paper.

######  Theorem 5.9.

Suppose (5.2) and that the local index-1 hypotheses (see Def. 4.14) are true
for $H-z_{0}W,W$ with respect to $J$ on the interval $\mathbb{R}$ for some
$z_{0}\in\mathbb{C}\setminus\mathbb{R}$. Then (5.3) and (5.4) are true, and

(5.24) $\displaystyle L:D(L)\to L^{2}(\mathbb{R};W)\text{ is a self-adjoint
operator on }L^{2}(\mathbb{R};W),$ (5.25) $\displaystyle
L_{0}^{\prime}:D(L_{0}^{\prime})\to L^{2}(\mathbb{R};W)\text{ is essentially
self-adjoint with closure $L$ (i.e., $\overline{L_{0}^{\prime}}=L$)},$ (5.26)
$\displaystyle L\text{ has no eigenvalues of finite multiplicity}.$

###### Proof.

Assume the hypotheses. First, we have that

(5.27) $\displaystyle(H-z_{0}W)^{*}=H-\overline{z_{0}}W.$

Thus, the local index-1 hypotheses are true for both $H-z_{0}W,W$ and
$(H-z_{0}W)^{*},W$ with respect to $J$ on the interval $\mathbb{R}$ (by
Corollary 4.16). Second,

(5.28) $\displaystyle\mathcal{L}-z_{0}I$
$\displaystyle=\mathcal{L}_{H-z_{0}},L-z_{0}I=L_{H-z_{0}W},L_{0}^{\prime}-z_{0}I=L_{H-z_{0}W,0}^{\prime},$
(5.29) $\displaystyle\mathcal{L}-\overline{z_{0}}I$
$\displaystyle=\mathcal{L}_{H-\overline{z_{0}}W},L-\overline{z_{0}}I=L_{H-\overline{z_{0}}W},L_{0}^{\prime}-\overline{z_{0}}I=L_{H-\overline{z_{0}}W,0}^{\prime}.$

Thus, by Theorem 4.26 it follows that the subspaces
$D(L_{H-z_{0}W,0}^{\prime})$, $D(L_{(H-z_{0}W)^{*},0}^{\prime})$,
$D(L_{H-z_{0}W}),$ and $D(L_{(H-z_{0}W)^{*}})$ are all dense in
$L^{2}(\mathbb{R};W)$,

(5.30)
$\displaystyle(L_{H-z_{0}W,0}^{\prime})^{*}=(\overline{L_{H-z_{0}W,0}^{\prime}})^{*}=L_{(H-z_{0}W)^{*}}=L_{H-\overline{z_{0}}W},$
(5.31) $\displaystyle
L_{(H-z_{0}W)^{*}}=\overline{L_{(H-z_{0}W)^{*}}}=(L_{(H-z_{0}W)^{*}})^{**},$
(5.32)
$\displaystyle(L_{H-z_{0}W})^{*}=(L_{(H-z_{0}W)^{*}})^{*}=\overline{L_{H-z_{0}W,0}^{\prime}}$

and

(5.33)
$\displaystyle(L_{H-\overline{z_{0}}W,0}^{\prime})^{*}=(L_{(H-z_{0}W)^{*},0}^{\prime})^{*}=(\overline{L_{(H-z_{0}W)^{*},0}^{\prime}})^{*}=L_{H-z_{0}W},$
(5.34) $\displaystyle
L_{H-z_{0}W}=\overline{L_{H-z_{0}W}}=(L_{H-z_{0}W})^{**},$ (5.35)
$\displaystyle(L_{H-z_{0}W})^{*}=\overline{L_{(H-z_{0}W)^{*},0}^{\prime}}.$

As we have

(5.36) $\displaystyle
D(L_{H-z_{0}W,0}^{\prime})=D(L_{0}-z_{0}I),\;D(L_{(H-z_{0}W)^{*},0}^{\prime})=D(L_{0}-\overline{z_{0}}I),$
(5.37) $\displaystyle
D(L_{H-z_{0}W})=D(L-z_{0}I),\;\;D(L_{(H-z_{0}W)^{*},0}^{\prime})=D(L-\overline{z_{0}}I)$

and

(5.38) $\displaystyle
D(L_{0}^{\prime}-z_{0}I)=D(L_{0}^{\prime}),\;D(L-z_{0}I)=D(L),$

it follows that the conclusions of Theorem 4.26, namely, statements (i), (ii),
and (iii) are all true for the interval $\mathbb{R}$. This implies that (5.3)
and (5.4) are true. Thus, by Theorem 5.8, in order to prove our theorem we
need only prove the claim

(5.39) $\displaystyle\dim\ker(\mathcal{L}-z_{0}I)<\infty\text{ and
}\dim\ker(\mathcal{L}-\overline{z_{0}}I)<\infty.$

If this was not true then either

(5.40) $\displaystyle\dim\ker(\mathcal{L}-z_{0}I)=\infty\text{ or
}\dim\ker(\mathcal{L}-\overline{z_{0}}I)=\infty$

and this would imply that

(5.41) $\displaystyle\dim\ker(\mathcal{L}_{[0,d]}-z_{0}I)=\infty\text{ or
}\dim\ker(\mathcal{L}_{[0,d]}-\overline{z_{0}}I)=\infty$

(where $\mathcal{L}_{[0,d]}$ denotes the DA operator associated with
$[0,d],J,H|_{[0,d]},W|_{[0,d]}$), a contradiction of Corollary 4.18. This
proves the claim. Therefore, our theorem has been proven. ∎

Using Theorem 5.9, we now give an example of a maximal operator $L$ that is
self-adjoint, has no eigenvalues of finite multiplicity, but does have an
eigenvalue of infinite multiplicity.

######  Example 5.10.

Consider the $d$-periodic DAEs (for any $d>0$) in canonical form:

$J\frac{d}{dt}f+Hf=\lambda Wf,$

where

$\displaystyle J=i\begin{bmatrix}1&0\\\
0&0\end{bmatrix},\;H=\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\;W=\begin{bmatrix}1&0\\\
0&1\end{bmatrix}=I_{2},\;f=\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix},$

that is,

$\displaystyle i\begin{bmatrix}1&0\\\
0&0\end{bmatrix}\frac{d}{dt}\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix}=\lambda\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}.$

Then the DA operator
$\mathcal{L}:D(\mathcal{L})\to[\mathcal{M}(\mathbb{R})]^{2}$ associated with
$\mathbb{R},J,H,W$ is

$\displaystyle
D(\mathcal{L})=\left\\{f\in[\mathcal{M}(\mathbb{R})]^{2}:Jf\in[W^{1,1}_{loc}(\mathbb{R})]^{2}\right\\}=\left\\{\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix}:f_{1}\in
W^{1,1}_{loc}(\mathbb{R}),f_{2}\in\mathcal{M}(\mathbb{R})\right\\},$
$\displaystyle\mathcal{L}f=W^{-1}\left(\frac{d}{dt}Jf+Hf\right)=\begin{bmatrix}i\frac{df_{1}}{dt}\\\
0\end{bmatrix},\;\;f=\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}\in
D(\mathcal{L}).$

Next, the Hilbert space $L^{2}(\mathbb{R};W)$ is just

$\displaystyle L^{2}(\mathbb{R};W)=[L^{2}(\mathbb{R})]^{2},$

and the maximal and minimal operators, $L:D(L)\rightarrow L^{2}(\mathbb{R};W)$
and $L_{0}^{\prime}:D(L_{0}^{\prime})\rightarrow L^{2}(\mathbb{R};W)$,
generated by $\mathcal{L}$ are

$\displaystyle Lf=\mathcal{L}f=\begin{bmatrix}i\frac{df_{1}}{dt}\\\
0\end{bmatrix},\;\;\text{for }f=\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}\in
D(L),$
$\displaystyle\;L_{0}^{\prime}f=\mathcal{L}f=\begin{bmatrix}i\frac{df_{1}}{dt}\\\
0\end{bmatrix},\;\;\text{for }f=\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}\in
D(L_{0}^{\prime}),$

where

$\displaystyle D(L)=\\{f\in L^{2}(\mathbb{R};W):f\in
D(\mathcal{L}),\mathcal{L}f\in L^{2}(\mathbb{R};W)\\}$
$\displaystyle=\left\\{\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}:f_{1}\in
W^{1,1}_{loc}(\mathbb{R}),f_{1},\frac{df_{1}}{dt},f_{2}\in
L^{2}(\mathbb{R})\right\\},$ $\displaystyle D(L_{0}^{\prime})=\\{f\in
D(L):Jf\text{ has compact support contained in the interior of }\mathbb{R}\\}$
$\displaystyle=\left\\{\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}:f_{2}\in
L^{2}(\mathbb{R}),f_{1}\in W^{1,1}_{loc}(\mathbb{R}),f_{1}\text{ has compact
support}\right\\}.$

Let us now prove that $L$ has only one eigenvalue, namely, $\lambda=0$, and it
is an eigenvalue of infinite multiplicity. Let $\lambda\in\mathbb{C}$,
$\lambda\not=0$. Then

$\displaystyle\mathcal{L}f=\lambda f\Leftrightarrow f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},f_{2}=0,f_{1}\in
W^{1,1}_{loc}(\mathbb{R}),\frac{df_{1}}{dt}=-i\lambda f_{1}$
$\displaystyle\Leftrightarrow f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},f_{2}=0,f_{1}(t)=ce^{-i\lambda t},c\in\mathbb{C}$

from which it follows that

$\displaystyle Lf=\lambda f\Leftrightarrow f=0.$

This proves that if $\lambda\not=0$ then $\lambda$ is not an eigenvalue of
$L$. Next, we will prove $\lambda=0$ is an eigenvalue of $L$ of infinite
multiplicity. First,

$\displaystyle\mathcal{L}f=0\Leftrightarrow f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},f_{2}\in\mathcal{M}(\mathbb{R}),f_{1}\in
W^{1,1}_{loc}(\mathbb{R}),\frac{df_{1}}{dt}=0$ $\displaystyle\Leftrightarrow
f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},f_{2}\in\mathcal{M}(\mathbb{R}),f_{1}(t)=c,c\in\mathbb{C}\text{
(i.e., $f_{1}$ is a constant function)}$

from which it follows that

$\displaystyle Lf=0\Leftrightarrow f=\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix},f_{2},f_{1}\in
L^{2}(\mathbb{R}),f_{1}(t)=c,c\in\mathbb{C}\Leftrightarrow\begin{bmatrix}0\\\
f_{2}\end{bmatrix},f_{2}\in L^{2}(\mathbb{R}),$

which proves that $\lambda=0$ is an eigenvalue of $L$ with infinite
multiplicity since its eigenspace $E_{0}$, i.e.,

$\displaystyle E_{0}=\left\\{\begin{bmatrix}0\\\ f_{2}\end{bmatrix}:f_{2}\in
L^{2}(\mathbb{R})\right\\},$

is infinite dimensional.

Our final goal here is to prove that $L$ is self-adjoint by proving that the
hypotheses of Theorem 5.9 are satisfied (which also then shows that
$L_{0}^{\prime}$ is essentially self-adjoint with closure $L$). To do this,
let $z_{0}\in\mathbb{C}$. We now want to consider whether or not the local
index-$1$ hypotheses (see Def. 4.14) are true for $H-z_{0}W,W$ with respect to
$J$ on the interval $\mathbb{R}$, where in this example

$\displaystyle H-z_{0}W=-z_{0}I_{2}.$

First, the unitary matrix

$V=I_{2}\in M_{2}(\mathbb{C}),$

already has

$J=V^{-1}JV=[J_{ij}]_{i,j=1,2}$

in $2\times 2$ block partitioned matrix form as (4.28) with

$\displaystyle J_{ij}\in M_{n_{i}\times
n_{j}}(\mathbb{C}),i,j=1,2;\;\;J_{11}=iI_{1},\;J_{ij}=0,\;\;(i,j)\not=(1,1),\;\;\det(J_{11})\neq
0,$

where $I_{1}=[1]$ is the $1\times 1$ identity matrix and
$n_{1},n_{2}\in\mathbb{N}$ are defined by

$\displaystyle n_{1}:=\operatorname{rank}J=1,\;\;n_{2}:=\dim\ker
J=\operatorname{nullity}(J)=2-n_{1}=1.$

Furthermore, both of the matrices
$H=V^{-1}HV=[H_{ij}]_{i,j=1,2},W=V^{-1}WV=[W_{ij}]_{i,j=1,2}\in
M_{2}(\mathcal{M}(\mathbb{R}))$ are block partitioned already conformal to the
block structure of $V^{-1}JV$ in (4.28), where

$\displaystyle H_{ij}=0,W_{ij}\in M_{n_{i}\times
n_{j}}(\mathcal{M}(\mathbb{R})),\;\;i,j=1,2;\;W_{11}=W_{22}=I_{1},\;W_{12}=W_{21}=0.$

Next, it follows from this that the local index-$1$ hypotheses (see Def. 4.14)
for $H-z_{0}W,W$ with respect to $J$ on the interval $\mathbb{R}$ are not
satisfied if $z_{0}=0$ since $H_{22}=0$ is not invertible, but are satisfied
if $z_{0}\not=0$ since

$\displaystyle\frac{-1}{z_{0}}I_{1}=(H-z_{0}W)_{22}^{-1}\in
M_{n_{2}}(\mathcal{M}(\mathbb{R})),$ $\displaystyle
0=(H-z_{0}W)_{12}(H-z_{0}W)_{22}^{-1}W_{22}^{1/2}\in M_{n_{1}\times
n_{2}}(L^{2}_{loc}(\mathbb{R})),$ $\displaystyle-
z_{0}I_{1}=(H-z_{0}W)/(H-z_{0}W)_{22},W_{11}=I_{1}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R})),$
$\displaystyle\frac{-1}{z_{0}}I_{1}=W_{22}^{1/2}(H-z_{0}W)_{22}^{-1}W_{22}^{1/2}\in
M_{n_{2}}(L^{\infty}_{loc}(\mathbb{R})),$ $\displaystyle
0=W_{22}^{1/2}(W_{22}^{-1}W_{21}-H_{22}^{-1}H_{21})\in M_{n_{2}\times
n_{1}}(L^{2}_{loc}(\mathbb{R})),$ $\displaystyle I_{1}=W/W_{22}\in
M_{n_{1}}(L^{1}_{loc}(\mathbb{R})).$

It follows from this that the local index-$1$ hypotheses (see Def. 4.14) for
$H-z_{0}W,W$ with respect to $J$ on the interval $\mathbb{R}$ are true in this
example if and only if $z_{0}\in\mathbb{C}\setminus\\{0\\}$. In particular, we
can take $z_{0}=i$ from which it follows in this example that $L$ is self-
adjoint by Theorem 5.9, has $\lambda=0$ as its only eigenvalue, and this
eigenvalue has infinite multiplicity with eigenspace $E_{0}$ above.

###  5.1. Simplifying the local index-1 hypotheses

As Theorem 5.9 is the main result of this paper, we would like to find simpler
hypotheses that are sufficient for those hypotheses in Theorem 5.9 to be true.
The next proposition and the theorem that follows are our main results in this
regard. First, we begin with a lemma.

######  Lemma 5.11.

If (5.2) is satisfied and $z_{0}\in\mathbb{C}\setminus\mathbb{R}$, then
$(W^{-1/2}HW^{-1/2}-z_{0}I_{n})^{-1}$ and $H-z_{0}W$ are invertible for a.e.
$t\in\mathbb{R}$ and

(5.42) $\displaystyle
W^{1/2}(H-z_{0}W)^{-1}W^{1/2}=(W^{-1/2}HW^{-1/2}-z_{0}I_{n})^{-1}\in
M_{n}(L^{\infty}_{loc}(\mathbb{R})),$ (5.43)
$\displaystyle(H-z_{0}W)^{-1}=W^{-1/2}(W^{-1/2}HW^{-1/2}-z_{0}I_{n})^{-1}W^{-1/2}\in
M_{n}(\mathcal{M}(\mathbb{R})).$

Furthermore, if $\det J=0$ then $(W^{-1/2}HW^{-1/2}-z_{0}I_{n})^{-1}$ and
$(H-z_{0}W)_{22}$ are invertible for a.e. $t\in\mathbb{R}$ and

(5.44) $\displaystyle
W_{22}^{1/2}(H-z_{0}W)_{22}^{-1}W_{22}^{1/2}=(W_{22}^{-1/2}H_{22}W_{22}^{-1/2}-z_{0}I_{n_{2}})^{-1}\in
M_{n_{2}}(L^{\infty}_{loc}(\mathbb{R})),$ (5.45)
$\displaystyle(H-z_{0}W)_{22}^{-1}=(H_{22}-z_{0}W_{22})^{-1}$
$\displaystyle=W_{22}^{-1/2}(W_{22}^{-1/2}H_{22}W_{22}^{-1/2}-z_{0}I_{n_{2}})^{-1}W_{22}^{-1/2}\in
M_{n_{2}}(\mathcal{M}(\mathbb{R})).$

###### Proof.

Suppose (5.2) is satisfied. Let $z_{0}\in\mathbb{C}\setminus\mathbb{R}$. Now
recall, for any $A\in M_{m}(\mathbb{C})$ such that $A^{*}=A$, it follows that
for any $x\in\mathbb{C}^{m}$,

$\displaystyle||(A-z_{0}I_{m})x||||x||\geq|\langle
x,(A-z_{0}I_{m})x\rangle|\geq|\operatorname{Im}\langle
x,(A-z_{0}I_{m})x\rangle|$ $\displaystyle=|\langle
x,\operatorname{Im}(A-z_{0}I_{m})x\rangle|=|\operatorname{Im}(z_{0})||x||^{2}$

which simultaneously proves that $A-z_{0}I_{m}$ is invertible and that

$\displaystyle||(A-z_{0}I_{m})^{-1}||\leq\frac{1}{|\operatorname{Im}(z_{0})|}.$

Next, if $C,B\in M_{m}(\mathbb{C})$ such that $C^{*}=C$ and $B$ is positive
semidefinite and invertible then $A:=B^{-1/2}CB^{-1/2}$ satisfies $A^{*}=A$
and

(5.46) $\displaystyle
C-z_{0}B=B^{1/2}(B^{-1/2}CB^{-1/2}-z_{0}I_{m})B^{1/2}=B^{1/2}(A-z_{0}I_{m})B^{1/2}$

so that $C-z_{0}B$ is invertible with

(5.47)
$\displaystyle(C-z_{0}B)^{-1}=B^{-1/2}(A-z_{0}I_{m})^{-1}B^{-1/2},\;(A-z_{0}I_{m})^{-1}=B^{1/2}(C-z_{0}B)^{-1}B^{1/2}.$

Finally, recall that if $D\in M_{m}(\mathbb{C})$ is any invertible matrix
then, in terms of its determinant $\det D$ and its adjugate
$\operatorname{adj}D$ (i.e., the transpose of the cofactor matrix of $D$), the
inverse formula holds:

$\displaystyle D^{-1}=\frac{1}{\det D}\operatorname{adj}D.$

The proof of this lemma now follows immediately from these elementary facts
and the hypotheses on $H,W$ along with the equality
$(H-z_{0}W)_{22}=H_{22}-z_{0}W_{22}$ in $M_{n_{2}}(\mathcal{M}(\mathbb{R}))$
in the case $\det J=0$. ∎

The next proposition gives simpler conditions, which are equivalent to the
hypotheses of Theorem 5.9.

######  Proposition 5.12.

Suppose (5.2) is satisfied. Then the following statements are true:

  * (a)

If $\det J\not=0$ then $H,W\in M_{n}(L^{1}_{loc}(\mathbb{R}))$ if and only if
the local index-1 hypotheses (see Def. 4.14) are true for $H-z_{0}W,W$ with
respect to $J$ on the interval $\mathbb{R}$, for some