Aref Jeribi
Spectral Theory and
Applications of Linear
Operators and Block
Operator Matrices
Spectral Theory and Applications of Linear
Operators and Block Operator Matrices
Aref Jeribi
Spectral Theory
and Applications of Linear
Operators and Block
Operator Matrices
123
Aref Jeribi
Department of Mathematics
University of Sfax
Sfax, Tunisia
ISBN 978-3-319-17565-2
ISBN 978-3-319-17566-9 (eBook)
DOI 10.1007/978-3-319-17566-9
Library of Congress Control Number: 2015102923
Springer Cham Heidelberg New York Dordrecht London
© Springer International Publishing Switzerland 2015
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To
my mother Sania,
my father Ali,
my wife Fadoua,
my children Adam and Rahma,
my brothers Sofien and Mohamed Amin,
my sister Elhem,
my mother-in-law Zineb,
my father-in-law Ridha, and
all members of my extended family
Preface
Several books have been devoted to the spectral theory and its applications.
However, this volume is very special as compared with the previous ones. For
example, the perturbation approach has been, for a long time, extensively studied
and considered as one of the most useful methods used in order to study some
mathematical and applied problems.
The main idea is that, if we know something about the solution for an easier
problem, lying “close” to the one we are studying, then we can say something about
our problem, provided that the difference or the perturbation is sufficiently weak.
In view of more advanced applications, especially the ones dealing with complicated evolutional problems in Physics, Chemistry, Technology, Biology, etc.,
where the natural setting doesn’t involve single operators but operator matrices
and polynomial operator pencils, the concept of compact perturbations is very often
used, and it was shown that they were not sufficient for handling such problems. The
main advantage of this book is the detailed description of the ways showing how the
compactness condition can be relaxed, in a very general Banach space setting, so
that the previously impossible problems become suddenly solvable. The method of
extending results is not unique. That is why we have to devote a lot of space in order
to describe the different extensions of the classical notions, and to demonstrate how
they specifically work in different applications.
More precisely, it is well known that the essential spectrum of an operator A
consists of those points of the spectrum which cannot be removed from the spectrum
by the addition to A of a compact operator. The most powerful result obtained in my
thesis is that, in L1 -spaces, the essential spectrum of an operator A is nothing else
but the largest subset of the spectrum of A which remains invariant under weakly
compact perturbations of A. This unexpected result has opened many prospects to
develop innovative ways leading to a rigorous study of the Fredholm theory and in
the whole book, we give an account of the recent research on the spectral theory
by presenting a wide panorama of techniques including the weak topology, which
vii
viii
Preface
contributes to an extra insight to the classical results and enables us to solve concrete
problems from transport theory arising in their natural setting (L1 -spaces). The main
topics include:
• Riesz theory of polynomially compact operators.
• Time behavior of solutions for an abstract Cauchy problem on Banach spaces.
• Fredholm theory and characterization of essential spectra by means of measure
of noncompactness, demicompact operator, measure of weak noncompactness,
and graph measures.
• S -essential spectra and essential pseudospectra.
• Spectral theory of block operator matrices.
• Spectral graph theory.
• Applications in mathematical physics and biology.
We do hope that this book will be very useful for researchers, since it represents
not only a collection of a previously heterogeneous material, but also an innovation
through several extensions.
Of course, it is impossible for a single book to cover such a huge field of
research. In making personal choices for inclusion of material, we tried to give
useful complementary references in this research area, hence probably neglecting
some relevant works. We would be very grateful to receive any comments from
readers and researchers, providing us with some information concerning some
missing references.
We would like to thank Salma Charfi for the improvement she has made in
the introduction of this book. So, we are indebted to her. We would like to thank
Nedra Moalla for the improvements she has made concerning the spectral mapping
theorem. We would also like to thank Aymen Ammar for the improvements he
has made throughout this book. So, we are very grateful to him. Concerning the
chapter related to graph theory, we were fortunate to have the help of Hatem
Baloudi, who assisted in the preparation of this chapter. So, we are indebted to
him. We would like to thank Professor Sylvain Golénia for his generous permission
to integrate, in this book, the results of Hatem Baloudi dealing with the graph
theory. Moreover, we would like to mention that the thesis work results, performed
under my direction, by my former students and presently colleagues Nedra Moalla,
Afif Ben Amar, Faiçal Abdmouleh, Boulbeba Abdelmoumen, Salma Charfi, Ines
Walha, Bilel Krichen, Omar Jedidi, Sonia Yengui, Aymen Ammar, Naouel Ben Ali,
Rihab Moalla, Hatem Baloudi, Mohammed Zerai Dhahri, and Bilel Boukettaya, the
obtained results have helped us in writing this book. Last but not least, we would like
to thank Ridha Damak for improving the English of all chapters of this book. Finally,
we apologize in case we have forgotten to quote any author who has contributed,
directly or indirectly, to this work.
Sfax, Tunisia
June 2015
Aref Jeribi
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Spectral Theory and Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Time Behavior of Solutions to an Abstract Cauchy
Problem on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Fredholm Theory and Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
S -Essential Spectra and Essential Pseudospectra . . . . . . . . . . . . . . . . .
1.5
Spectral Theory of Block Operator Matrices . . . . . . . . . . . . . . . . . . . . . .
1.6
Spectral Graph Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7
Applications in Mathematical Physics and Biology. . . . . . . . . . . . . . .
1.8
Outline of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1
Closed and Closable Operators. . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2
Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3
Elementary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.4
Fredholm Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.5
Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.6
Relatively Boundedness and Relatively
Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.7
Sum of Closed Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.8
Strictly Singular and Strictly Cosingular Operators . . . .
2.1.9
Fredholm and Semi-Fredholm Perturbations . . . . . . . . . . .
2.1.10
Dunford–Pettis Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
Basics on Bounded Fredholm Operators . . . . . . . . . . . . . . .
2.2.2
Gap Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3
Semi-Regular and Essentially Semi-Regular
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4
Basics on Unbounded Fredholm Operators. . . . . . . . . . . . .
1
2
4
8
13
15
16
18
20
23
23
23
24
25
26
28
29
31
32
33
35
36
37
42
44
48
ix
x
Contents
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
3
2.2.5
Quasi-Inverse Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6
Basics on Unbounded Browder Operators . . . . . . . . . . . . . .
Positive Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1
Positive Operator on Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2
Positive Operator on Banach Lattice . . . . . . . . . . . . . . . . . . . .
Integral Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1
Integral Operator on Lp -Spaces. . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2
Integral Operator on L1 -Spaces . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3
Cauchy’s Type Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Semigroup Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1
Strongly Continuous Semigroup . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2
The Hille-Yosida Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Essential Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Borel Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Baire Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Banach Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measure of Noncompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.1
Measure of Noncompactness of a Bounded Subset . . . .
2.10.2
Measure of Noncompactness of an Operator . . . . . . . . . . .
Measure of Weak Noncompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Graph Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12.1
Graph Measure of Noncompactness . . . . . . . . . . . . . . . . . . . .
2.12.2
Graph Measure of Weak Noncompactness . . . . . . . . . . . . .
2.12.3
Seminorm Related to Upper
Semi-Fredholm Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schur Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalities about graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15.1
Unoriented Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15.2
Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15.3
Bipartite Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15.4
Subgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fredholm Operators and Riesz Theory for Polynomially
Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Riesz Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1
Some Results on Polynomially Compact Operators . . .
3.1.2
Generalized Riesz Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
First and Second Kind Operator Equation . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Spectral Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Localization of Eigenvalues of Polynomially Compact
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Polynomially Riesz Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Some Results on Polynomially Fredholm Perturbation . . . . . . . . . . .
51
53
56
56
58
59
59
61
63
65
65
66
68
71
74
76
77
77
80
82
84
86
87
88
89
91
92
92
93
94
98
101
101
101
106
108
110
112
117
118
Contents
4
5
Time-Asymptotic Description of the Solution
for an Abstract Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Abstract Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1
Compactness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2
The Remainder Term of the Dyson–Philips
Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Time Behavior of Solutions for an Abstract Cauchy
Problem (4.0.1) on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Time Behavior of Solutions for an Abstract Cauchy
Problem (4.0.1) on Lp -Spaces (1 < p < 1) . . . . . . . . . . . . . . . . . . . . .
Fredholm Theory Related to Some Measures . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
Fredholm Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Fredholm Theory by Means of Noncompactness Measures . . . . . .
5.3
Fredholm Theory by Means of Non-strict
Singularity Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
Fredholm Theory by Means of Demicompact Operator . . . . . . . . . .
5.4.1
Demicompactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2
S -Demicompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
Fredholm Theory by Means of Weak Noncompactness
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6
Fredholm Theory with Finite Ascent and Descent . . . . . . . . . . . . . . . .
5.7
Stability of Semi-Browder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1
Convergence to Zero Compactly . . . . . . . . . . . . . . . . . . . . . . . .
xi
121
121
122
124
127
136
139
139
145
148
150
150
155
158
164
165
167
6
Perturbation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Fredholm and Semi-Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Semi-Fredholm Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Fredholm Inverse Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
Fredholm Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6
Some Perturbation Results for Matrix Operators . . . . . . . . . . . . . . . . . .
6.7
Some Fredholm Theory Results for Matrix Operators . . . . . . . . . . . .
173
173
174
175
181
185
187
191
7
Essential Spectra of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
The Jeribi Essential Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1
Relationship Between Jeribi and Schechter
Essential Spectra on L1 -Spaces . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2
Relationship Between Jeribi and Schechter
Essential Spectra on Banach Space
Satisfying the Dunford–Pettis Property . . . . . . . . . . . . . . . . .
7.2.3
Other Characterization of the Schechter
Essential Spectrum by the Jeribi Essential
Spectrum on Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
193
193
198
199
199
200
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Contents
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Essential Spectra of the Sum of Two Bounded Linear
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1
By Means of Fredholm and Semi-Fredholm
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.2
By Means of Fredholm Inverse . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.3
By Means of Demicompact Operators . . . . . . . . . . . . . . . . . .
Unbounded Linear Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1
Essential Spectra for the Sum of Closed
and Bounded Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.2
Essential Spectra for the Product of Closed
and Bounded Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.3
Invariance of the Essential Spectra. . . . . . . . . . . . . . . . . . . . . .
7.5.4
Characterization of the Rakoc̆ević and
Schmoeger Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Invariance of the Kato Spectrum by Commuting
Nilpotent Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Invariance of Schechter’s Essential Spectrum . . . . . . . . . . . . . . . . . . . . .
7.7.1
Characterization of Schechter’s Essential Spectrum . . .
7.7.2
Invariance by Means of Demicompact Operators . . . . . .
7.7.3
Invariance by Means of Noncompactness Measure . . . .
7.7.4
Invariance of the Schechter’s Essential
Spectrum in Dunford–Pettis Space. . . . . . . . . . . . . . . . . . . . . .
7.7.5
Invariance Under Perturbation of
Polynomially Compact Operators . . . . . . . . . . . . . . . . . . . . . . .
7.7.6
Invariance by Means of Weak
Noncompactness Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stability of the Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.1
By Means of Measure of Weak Noncompactness . . . . . .
7.8.2
By Means of the Graph Measure of Weak
Noncompactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.3
By Measure of Non-upper Semi-Fredholm
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.4
Generalized Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.5
Convergence to Zero Compactly . . . . . . . . . . . . . . . . . . . . . . . .
Borel Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectral Mapping Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Characterization of Polynomially Riesz Strongly
Continuous Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.11.1
Polynomially Fredholm Perturbations . . . . . . . . . . . . . . . . . .
7.11.2
Polynomially Riesz Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Spectral Characterization of the Uniform
Continuity of Strongly Continuous Groups. . . . . . . . . . . . . . . . . . . . . . . .
200
204
204
212
214
216
216
218
221
227
229
231
231
234
236
237
241
245
249
249
250
252
253
257
260
265
267
267
268
274
Contents
7.13
8
9
10
xiii
Some Results on Strongly Continuous Semigroups
of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
7.13.1
Arbitrary Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
7.13.2
Hereditarily Indecomposable Banach Spaces . . . . . . . . . . 280
Pseudo-Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1
Pseudo-Spectrum of Linear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1
Approximation of "-Pseudo-Spectrum. . . . . . . . . . . . . . . . . .
8.1.2
Approximation of the Spectrum. . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Pseudo-Browder’s Essential Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1
Extending the Resolvent to a Browder
Resolvent b .:/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.2
Definition of the Pseudo-Browder Essential
Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.3
Pseudo-Browder’s Essential Spectrum
of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.4
Characterization of the Pseudo-Browder
Essential Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.5
Stability of the Pseudo-Browder’s Essential
Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3
Pseudo-Jeribi and Pseudo-Schechter Essential Spectra . . . . . . . . . . .
8.3.1
By Means of a Fredholm and
Semi-Fredholm Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2
By Means of Noncompactness Measure . . . . . . . . . . . . . . . .
283
283
285
288
293
S -Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1
Definitions and Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Characterization of S-Essential Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3
The S -Browder’s Essential Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4
S -Essential Spectra of the Sum of Bounded Linear Operators . . .
9.5
S -Essential Spectra by Means of Demicompact Operators . . . . . . .
9.6
Characterization of the Relative Schechter’s
and Approximate Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
309
309
311
313
319
323
Essential Spectra of 2 2 Block Operator Matrices. . . . . . . . . . . . . . . . . . . .
10.1
Case Where the Resolvent of the Operator A
Is a Fredholm Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.1
Frobenius–Schur’s Decomposition . . . . . . . . . . . . . . . . . . . . .
10.1.2
Closability and Closure of the Block
Operator Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.3
Essential Spectra of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.4
Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2
Case Where the Operator A Is Closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1
Closability and Closure of the Block
Operator Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
327
293
295
296
297
299
300
301
306
325
328
328
332
333
336
338
339
xiv
Contents
10.3
10.4
10.5
10.6
10.7
11
12
10.2.2
Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.3
Particular Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Case Where the Operator A Is Closable . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1
Closability and Closure of the Block
Operator Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2
Essential Spectra of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.3
The Operator A as an Infinitesimal
Generator of a Holomorphic Semigroup. . . . . . . . . . . . . . . .
Relative Boundedness for Block Operator Matrices . . . . . . . . . . . . . .
Stability of the Wolf Essential Spectrum of Some
Matrix Operators Acting in Friedrichs Module. . . . . . . . . . . . . . . . . . . .
Wolf Essential Spectrum of Block Operator Matrix
Acting in Friedrichs Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The M -Essential Spectra of Block Operator Matrices . . . . . . . . . . . .
10.7.1
Closability and Closure of the Block
Operator Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.7.2
Essential Spectra of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
340
343
346
347
351
360
362
367
369
371
372
373
Essential Spectra of 3 3 Block Operator Matrices. . . . . . . . . . . . . . . . . . . .
11.1
Case Where the Operator A Is Closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.1
The Operator L0 and Its Closure . . . . . . . . . . . . . . . . . . . . . . . .
11.1.2
Essential Spectra of the Operator L . . . . . . . . . . . . . . . . . . . . .
11.2
Case Where the Operator A Is Closable . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1
Closability and Closure of the Operator L0 . . . . . . . . . . . .
11.2.2
Gustafson, Weidman, Kato, Wolf,
Schechter, Browder, Rakočević, and
Schmoeger’s Essential Spectra of L . . . . . . . . . . . . . . . . . . . .
11.3
Block Operator Matrices Using Browder Resolvent . . . . . . . . . . . . . .
11.3.1
The Operator A0 and Its Closure. . . . . . . . . . . . . . . . . . . . . . . .
11.3.2
Rakočević and Schmoeger Essential
Spectra of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4
Perturbations of Unbounded Fredholm Linear Operators. . . . . . . . .
11.4.1
The Operator A and Its Closure . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.2
Index of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
405
407
407
410
Spectral Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1
Line graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2
Operators on Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3
Lower Local Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4
On the Persson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.1
Spectral Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2
Persson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5
Essential Self-Adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.1
Unbounded Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.2
A Counter Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
413
413
415
416
421
421
421
423
423
425
375
375
375
379
381
382
392
398
398
Contents
12.5.3
Nelson Commutator Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5.4
Application on Schur Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Adjacency Matrix on Line Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6.1
Oriented Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6.2
Case of Bipartite Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
426
429
430
430
434
Applications in Mathematical Physics and Biology . . . . . . . . . . . . . . . . . . . . .
13.1
Time-Asymptotic Description of the Solution
for a Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.1
Preliminaries and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.2
Compactness and Generation Results. . . . . . . . . . . . . . . . . . .
13.1.3
Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.4
Solution for the Cauchy Problem (13.1.1) . . . . . . . . . . . . . .
13.1.5
Generation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.6
Time-Asymptotic Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2
Time-Asymptotic Behavior of the Solution
for a Cauchy Problem Given by a One-Velocity
Transport Operator with Maxwell Boundary Condition . . . . . . . . . .
13.2.1
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.2
The Resolvent of the Operator B˛ . . . . . . . . . . . . . . . . . . . . . .
13.2.3
Generation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2.4
Asymptotic Behavior of the Solution . . . . . . . . . . . . . . . . . . .
13.3
The Time-Asymptotic Behavior of a Transport
Operator with a Diffuse Reflection Boundary Condition . . . . . . . . .
13.3.1
The Resolvent of the Operator B . . . . . . . . . . . . . . . . . . . . . . .
13.3.2
Compactness and Generation Results. . . . . . . . . . . . . . . . . . .
13.3.3
Asymptotic Behavior of the Solution . . . . . . . . . . . . . . . . . . .
13.4
Essential Spectra of Transport Operator Arising
in Growing Cell Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.1
The Resolvent of the Operator SK . . . . . . . . . . . . . . . . . . . . . .
13.4.2
Spectral Properties of SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.3
Generation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.4
Compactness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.5
The Irreducibility of the Semigroup e tAk . . . . . . . . . . . . . . .
13.4.6
Existence of the Leading Eigenvalues of AK . . . . . . . . . . .
13.4.7
The Strict Monotonicity of the Leading
Eigenvalue of AK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.8
Essential Spectra of AK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5
Some Applications of the Regularity and
Irreducibility on Transport Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.1
Weak Compactness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.2
Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.3
Existence Results of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . .
13.5.4
Monotonicity of the Spectral Bound . . . . . . . . . . . . . . . . . . . .
13.6
Singular Neutron Transport Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
441
12.6
13
xv
441
442
446
450
464
464
465
469
470
473
475
477
481
484
487
491
496
496
498
505
508
510
512
515
517
519
522
526
527
532
533
xvi
Contents
13.7
13.8
13.9
13.10
13.11
13.12
13.13
Systems of Ordinary Differential Operators . . . . . . . . . . . . . . . . . . . . . . .
Essential Spectra of Two-Group Transport Operators . . . . . . . . . . . .
13.8.1
The Expression of the Resolvent of TH1 . . . . . . . . . . . . . . . .
13.8.2
Compactness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8.3
Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Elliptic Problems with -Dependent Boundary Conditions . . . . . .
13.9.1
The Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9.2
Verification of the Assumptions .J1/–.J 8/
of Chap. 10, Sect. 10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9.3
The Closure of the Operator A0 . . . . . . . . . . . . . . . . . . . . . . . . .
13.9.4
Spectrum of the Operator A . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9.5
Semigroup Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A -Rational Sturm–Liouville’s Problem . . . . . . . . . . . . . . . . . . . . . . . . .
Two-Group Radiative Transfer Equations in a Channel. . . . . . . . . . .
13.12.1 Functional Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.12.2 The Expression of the Resolvent
of the Operator T2H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.12.3 Essential Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Three-Group Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
537
544
545
547
550
553
553
555
556
557
558
559
561
563
563
566
568
573
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
Chapter 1
Introduction
This book is devoted to some recent mathematical developments which cover
several topics including Cauchy problem, Fredholm operators, spectral theory,
and block operator matrices, both dealing with linear operators. Of course, these
topics play a crucial role in many branches of mathematics and also in numerous
applications as they are intimately related to the stability of the underlying physical
systems.
One of the objectives of this book is the study of the classical Riesz theory
of polynomially compact operators, in order to establish the existence results of
the second kind operator equations, hence allowing to describe the spectrum,
multiplicities, and localization of the eigenvalues of polynomially compact operators. Fredholm theory and perturbation results are also widely investigated. The
description of the large time behavior of solutions to an abstract Cauchy problem on
Banach spaces without restriction on the initial data is studied. Further, the essential
state of the art of research and essential pseudo-spectra of closed, densely defined,
and linear operators subjected to additive perturbations is outlined. The spectral
theory of block operator matrices is of major interest, since it describes coupled
systems of partial differential equations of mixed order and type. For this reason,
an important part of this book is devoted to develop essential spectra of 2 2 and
3 3 block operator matrices. Based on the spectral graph theory (which is an active
research area), we are interested in the study of the adjacency matrix and the discrete
Laplacian acting on forms. Most of the results of this book are motivated by physical
transport problems for which we address our applications at the end of the book.
Now, let us describe its contents.
© Springer International Publishing Switzerland 2015
A. Jeribi, Spectral Theory and Applications of Linear Operators
and Block Operator Matrices, DOI 10.1007/978-3-319-17566-9_1
1
2
1 Introduction
1.1 Spectral Theory and Cauchy Problem
As it is well known for the second kind operator equations
' A' D f
(1.1.1)
in Banach spaces, the existence and uniqueness of a solution can be established by
the Neumann series, provided that A is a contraction, i.e., kAk < jj. The basic
theory for the second kind operator equation (1.1.1) with a compact linear operator
A on X was developed by F. Riesz [294] and originated by I. Fredholm’s work on
the second kind integral equations [113]. In [171, 254], A. Jeribi and N. Moalla
extended this analysis to the polynomially compact operator A in the more general
setting of normed spaces. Such an extension provided some solutions for several
physical problems. In fact, if A is a polynomially compact operator onP
a normed
p
r
space X , i.e., there exists a nonzero complex polynomial P .z/ D
rD0 ar z
satisfying P .A/ 2 K.X / (the set of compact operators), and if 2 C with
P ./ ¤ 0, then we have two cases:
if the homogeneous equation
' A' D 0
(1.1.2)
only has the trivial solution ' D 0 then, for all f 2 X , the non-homogeneous
equation (1.1.1) has a unique solution ' 2 X which depends continuously on f .
If the homogeneous equation (1.1.2) has a nontrivial solution, then the nonhomogeneous equation (1.1.1) is either unsolvable or its general solution is of the
following form
' D 'Q C
m
X
˛k 'k ;
kD1
where '1 ; : : : ; 'm are linearly independent solutions of the homogeneous equation,
˛1 ; : : : ; ˛m represent arbitrary complex numbers, and 'Q denotes a particular solution
of the non-homogeneous equation (1.1.1).
The structure of polynomially compact operators was described by F. Gilfeather
[117] and by Y. M. Han et al. [146] in the context of Hilbert spaces. F. Gilfeather
showed that every polynomially compact operator on a Banach space is the finite
direct sum of translates of operators which have the property that the finite power of
the operator is compact. Moreover, the spectrum of these operators can be described.
This analysis was widely developed by V. I. Istrateescu in [156].
It is well known that, if X is a complex Banach space, and if A 2 K.X /, then
A and A (the dual of A) are Riesz operators, with .A / D .A/ (see [191]).
Furthermore, N. Dunford and J. T. Schwartz showed in [101] that, for any eigenvalue
2 .A/nf0g, we have mult.A; / < 1 and mult.A; / D mult.A ; /, where
mult.:; :/ represents the algebraic multiplicity.
1.1 Spectral Theory and Cauchy Problem
3
Let us notice that, if A is a Riesz operator on X , then A is a generalized Riesz
operator on X . One of the purposes of the Chap. 2 is to prove that a polynomially
compact operator is also a generalized Riesz operator.
PpLet A r 2 PK.X /, i.e., there exists a nonzero complex polynomial P .z/ D
rD0 ar z satisfying P .A/ 2 K.X /. In [171], A. Jeribi and N. Moalla expressed the
multiplicity of a nonzero eigenvalue of P .A/ according to the one of the eigenvalues
of the operator A, and proved that, if B 2 L.Y / such that A and B are related
operators, then B is a generalized Riesz operator and mult.A; / D mult.B; /
for all 2 .A/nf0; z1 ; : : : ; zk g, where z1 ; : : : ; zk are the zeroes of the minimal
polynomial of A.
In [84], J. R. Cuthbert considered a class of C0 -semigroups .T .t //t0 satisfying
the property of being near the identity, which means that, for some values of t ,
T .t / I 2 K.X /. Cuthbert’s result asserts that, if .T .t //t0 is a C0 -semigroup with
an infinitesimal generator A, then the following conditions are equivalent:
(i) ft > 0 such that T .t / I is compactg D0; 1Œ,
(ii) A is compact, and
(iii) . A/1 I is compact for some (and then, for all) > w,
where w denotes the type of .T .t //t0 . Cuthbert’s result was extended by several
authors. Their aim was to study other strongly continuous families of operators such
as cosine or resolvent families of operators (see [151, 235, 240]). For example, in
the paper [218], the authors have shown that the assertions (i), (ii), and (iii) remain
equivalent for strongly continuous semigroups .T .t //t0 near the identity, which
explains the existence of t0 > 0, such that T .t0 /I 2 I.X /, where I.X / represents
any arbitrary, closed, and proper two-sided ideal of the algebra L.X / belonging to
F.X / (the set of Fredholm perturbations). Let us remark that, in all these works,
the generator A is either compact or belongs to an ideal of L.X / contained in
F.X /. The general case was considered in [155], where A was a Riesz operator,
not necessarily belonging to F.X /. We say that an operator A 2 L.X / belongs to
P I.X /, if there exists a nonzero complex polynomial p.:/, such that the operator
p.A/ 2 I.X /. In [225], the results obtained in [84, 155], and [218] were extended
to semigroups for which there exists a nontrivial polynomial p.:/ 2 CŒz such that,
for some t > 0, p.T .t // 2 I.X /. As opposed to the previous results, in this case,
the infinitesimal generator of the semigroup is not necessarily a Riesz operator.
In [229], the authors characterized the class of polynomially Riesz strongly
continuous semigroups on a Banach space X . In particular, their main results assert
that the generators of such semigroups are either polynomially Riesz (then bounded)
or there exist two closed, infinite-dimensional, and invariant subspaces X0 and X1
of X with X D X0 ˚X1 , such that the part of the generator in X0 is unbounded with
a resolvent of Riesz type, whereas its part in X1 is a polynomially Riesz operator.
In Chap. 2 of his thesis [351], M. Yahdi discussed the topological complexity of
some subsets of L.X /, under the assumptions that X is a separable Banach space
and L.X / is endowed with the strong operator topology. In particular, M. Yahdi
showed that the families of stable, ergodic, and power-bounded operators constitute