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E-raamat: Linear Operators and Their Essential Pseudospectra

(Point Pleasant, New Jersey, USA)
  • Formaat: 368 pages
  • Ilmumisaeg: 17-Apr-2018
  • Kirjastus: Apple Academic Press Inc.
  • ISBN-13: 9781351046251
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Linear Operators and Their Essential PseudospectraSuurem pilt
  • Formaat: 368 pages
  • Ilmumisaeg: 17-Apr-2018
  • Kirjastus: Apple Academic Press Inc.
  • ISBN-13: 9781351046251
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Linear Operators and Their Essential Pseudospectra provides a comprehensive study of spectral theory of linear operators defined on Banach spaces. The central items of interest in the volume include various essential spectra, but the author also considers some of the generalizations that have been studied.





In recent years, spectral theory has witnessed an explosive development. This volume presents a survey of results concerning various types of essential spectra and pseudospectra in a unified, axiomatic way and also discusses several topics that are new but which relate to the concepts and methods emanating from the book. The main topics include essential spectra, essential pseudospectra, structured essential pseudospectra, and their relative sets.





This volume will be very useful for several researchers since it represents not only a collection of previously heterogeneous material but also includes discussions of innovation through several extensions. As the spectral theory of operators is an important part of functional analysis and has numerous applications in many areas of mathematics, the author suggests that some modest prerequisites from functional analysis and operator theory should be in place to be accessible to newcomers and graduate students of mathematics.

About the Author xiii
Preface xv
1 Introduction
1(18)
1.1 Essential Spectra and Relative Essential Spectra
2(6)
1.2 Essential Pseudospectra
8(4)
1.3 Structured Essential Pseudospectra and Relative Structured Essential Pseudospectra
12(2)
1.4 Condition Pseudospectrum
14(1)
1.5 Outline of Contents
15(4)
2 Fundamentals
19(66)
2.1 Operators
19(15)
2.1.1 Linear Operators
19(1)
2.1.2 Bounded Operators
20(1)
2.1.3 Closed and Closable Operators
20(2)
2.1.4 Adjoint Operator
22(2)
2.1.5 Direct Sum
24(1)
2.1.6 Resolvent Set and Spectrum
25(1)
2.1.7 Compact Operators
26(1)
2.1.8 A-Defined, A-Bounded, and A-Compact Operators
27(2)
2.1.9 Weakly Compact and A-Weakly Compact Operators
29(1)
2.1.10 Dunford-Pettis Property
29(1)
2.1.11 Strictly Singular Operators
30(1)
2.1.12 Strictly Cosingular
31(1)
2.1.13 Perturbation Function
32(1)
2.1.14 Measure of Non-Strict-Singularity
33(1)
2.1.15 Semigroup Theory
33(1)
2.2 Fredholm and Semi-Fredholm Operators
34(14)
2.2.1 Definitions
34(3)
2.2.2 Basics on Bounded Fredholm Operators
37(6)
2.2.3 Basics on Unbounded Fredholm Operators
43(5)
2.3 Perturbation
48(9)
2.3.1 Fredholm and Semi-Fredholm Perturbations
48(3)
2.3.2 Semi-Fredholm Perturbations
51(1)
2.3.3 Riesz Operator
52(1)
2.3.4 Some Perturbation Results
53(1)
2.3.5 A-Fredholm Perturbation
54(1)
2.3.6 A-Compact Perturbations
55(1)
2.3.7 The Convergence Compactly
56(1)
2.4 Ascent and Descent Operators
57(6)
2.4.1 Bounded Operators
57(5)
2.4.2 Unbounded Operators
62(1)
2.5 Semi-Browder and Browder Operators
63(4)
2.5.1 Semi-Browder Operators
63(2)
2.5.2 Fredholm Operator with Finite Ascent and Descent
65(2)
2.6 Measure of Noncompactness
67(8)
2.6.1 Measure of Noncompactness of a Bounded Subset
68(1)
2.6.2 Measure of Noncompactness of an Operator
69(2)
2.6.3 Measure of Non-Strict-Singularity
71(2)
2.6.4 γ-Relatively Bounded
73(1)
2.6.5 Perturbation Result
73(2)
2.7 γ-Diagonally Dominant
75(1)
2.8 Gap Topology
76(3)
2.8.1 Gap Between Two Subsets
76(1)
2.8.2 Gap Between Two Operators
77(1)
2.8.3 Convergence in the Generalized Sense
78(1)
2.9 Quasi-Inverse Operator
79(4)
2.10 Limit Inferior and Superior
83(2)
3 Spectra
85(44)
3.1 Essential Spectra
85(6)
3.1.1 Definitions
85(3)
3.1.2 Characterization of Essential Spectra
88(3)
3.2 The Left and Right Jeribi Essential Spectra
91(1)
3.3 S-Resolvent Set, S-Spectra, and S-Essential Spectra
92(16)
3.3.1 The S-Resolvent Set
92(6)
3.3.2 S-Spectra
98(2)
3.3.3 S-Browder's Resolvent
100(4)
3.3.4 S-Essential Spectra
104(4)
3.4 Invariance of the S-Essential Spectrum
108(5)
3.5 Pseudospectra
113(11)
3.5.1 Pseudospectrum
113(2)
3.5.2 S-Pseudospectrum
115(4)
3.5.3 Ammar-Jeribi Essential Pseudospectrum
119(3)
3.5.4 Essential Pseudospectra
122(1)
3.5.5 Conditional Pseudospectrum
123(1)
3.6 Structured Pseudospectra
124(5)
3.6.1 Structured Pseudospectrum
124(1)
3.6.2 The Structured Essential Pseudospectra
125(1)
3.6.3 The Structured S-Pseudospectra
126(2)
3.6.4 The Structured S-Essential Pseudospectra
128(1)
4 Perturbation of Unbounded Linear Operators by γ-Relative Boundedness
129(14)
4.1 Sum of Closable Operators
129(8)
4.1.1 Norm Operators
129(6)
4.1.2 Kuratowski Measure of Noncompactness
135(2)
4.2 Block Operator Matrices
137(6)
4.2.1 2 × 2 Block Operator Matrices
137(2)
4.2.2 3 ×3 Block Operator Matrices
139(4)
5 Essential Spectra
143(40)
5.1 Characterization of Essential Spectra
143(13)
5.1.1 Characterization of Left and Right Weyl Essential Spectra
143(6)
5.1.2 Characterization of Left and Right Jeribi Essential Spectra
149(7)
5.2 Stability of Essential Approximate Point Spectrum and Essential Defect Spectrum of Linear Operator
156(6)
5.2.1 Stability of Essential Spectra
156(3)
5.2.2 Invariance of Essential Spectra
159(3)
5.3 Convergence
162(21)
5.3.1 Convergence Compactly
162(7)
5.3.2 Convergence in the Generalized Sense
169(14)
6 S-Essential Spectra of Closed Linear Operator on a Banach Space
183(22)
6.1 S-Essential Spectra
183(12)
6.1.1 Characterization of S-Essential Spectra
183(5)
6.1.2 Stability of S-Essential Spectra of Closed Linear Operator
188(7)
6.2 S-Left and S-Right Essential Spectra
195(10)
6.2.1 Stability of S-Left and S-Right Fredholm Spectra
195(6)
6.2.2 Stability of S-Left and S-Right Browder Spectra
201(4)
7 S-Essential Spectrum and Measure of Non-Strict-Singularity
205(12)
7.1 A Characterization of the S-Essential Spectrum
205(5)
7.2 The S-Essential Spectra of 2 × 2 Block Operator Matrices
210(7)
8 S-Pseudospectra and Structured S-Pseudospectra
217(22)
8.1 Study of the S-Pseudospectra
217(6)
8.2 Characterization of the Structured S-Pseudospectra
223(8)
8.3 Characterization of the Structured S-Essential Pseudospectra
231(8)
9 Structured Essential Pseudospectra
239(30)
9.1 On a Characterization of the Structured Wolf, Ammar-Jeribi, and Browder Essential Pseudospectra
239(23)
9.1.1 Structured Ammar-Jeribi, and Browder Essential Pseudospectra
239(13)
9.1.2 A Characterization of the Structured Browder Essential Pseudospectrum
252(10)
9.2 Some Description of the Structured Essential Pseudospectra
262(7)
9.2.1 Relationship Between Structured Jeribi and Structured Ammar-Jeribi Essential Pseudospectra
262(2)
9.2.2 A Characterization of the Structured Ammar-Jeribi Essential Pseudospectrum
264(5)
10 Structured Essential Pseudospectra and Measure of Noncompactness
269(10)
10.1 New Description of the Structured Essential Pseudospectra
269(10)
10.1.1 A Characterization of the Structured Ammar-Jeribi Essential Pseudospectrum by Kuratowski Measure of Noncompactness
269(5)
10.1.2 A Characterization of the Structured Browder Essential Pseudospectrum by Means of Measure of Non-Strict-Singularity
274(5)
11 A Characterization of the Essential Pseudospectra
279(36)
11.1 Approximation of ε-Pseudospectrum
279(4)
11.2 A Characterization of Approximation Pseudospectrum
283(10)
11.3 Essential Approximation Pseudospectrum
293(5)
11.4 Properties of Essential Pseudospectra
298(7)
11.5 Pseudospectrum of Block Operator Matrices
305(10)
12 Conditional Pseudospectra
315(12)
12.1 Some Properties of Σε(A)
315(7)
12.2 Characterization of Condition Pseudospectrum
322(5)
Bibliography 327(18)
Index 345
Aref Jeribi, PhD, is Professor in the Department of Mathematics at the University of Sfax, Tunisia. He is the author of the book Spectral Theory and Applications of Linear Operators and Block Operator Matrices (2015) and co-author of the book Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications (CRC Press, 2015). He has published many journal articles in international journals. His areas of interest include spectral theory, matrice operators, transport theory, Gribov operator, Bargman space, fixed point theory, Riesz basis, and linear relations.
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