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E-raamat: Bounds for Determinants of Linear Operators and their Applications [Taylor & Francis e-raamat]

(Ben Gurion University of the Negev, Israel)
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Bounds for Determinants of Linear Operators and their ApplicationsSuurem pilt
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This book deals with the determinants of linear operators in Euclidean, Hilbert and Banach spaces. Determinants of operators give us an important tool for solving linear equations and invertibility conditions for linear operators, enable us to describe the spectra, to evaluate the multiplicities of eigenvalues, etc. We derive upper and lower bounds, and perturbation results for determinants, and discuss applications of our theoretical results to spectrum perturbations, matrix equations, two parameter eigenvalue problems, as well as to differential, difference and functional-differential equations.

Preface ix
1 Preliminaries
1(20)
1.1 Inequalities for eigenvalues and singular numbers
1(2)
1.2 Inequalities for convex functions
3(1)
1.3 Perturbations of entire Banach valued functions
4(1)
1.4 Functions defined on quasi-normed spaces
5(2)
1.5 Upper bounds for Weierstrass factors
7(3)
1.6 Lower bounds for Weierstrass factors
10(3)
1.7 Perturbations of determinants of finite dimensional operators
13(2)
1.8 Proof of Theorem 1.7.1
15(1)
1.9 Matrices with dominant principal diagonals
16(1)
1.10 Additional inequalities for determinants of matrices
17(2)
1.11 Comments to
Chapter 1
19(2)
2 Determinants of Schatten-von Neumann Operators
21(20)
2.1 Schatten-von Neumann ideals
21(2)
2.2 Examples of Hilbert-Schmidt and nuclear operators
23(4)
2.2.1 Infinite matrices
23(2)
2.2.2 Integral operators
25(2)
2.3 The characteristic determinant of a nuclear operator
27(3)
2.4 Regularized determinants of Schatten-von Neumann operators
30(1)
2.5 Upper bounds for regularized determinants
31(3)
2.6 Lower bounds for regularized determinants
34(1)
2.7 Perturbations of determinants of Schatten-von Neumann operators
35(3)
2.8 Invertibility of infinite matrices
38(1)
2.9 Comments to
Chapter 2
39(2)
3 Determinants of Nakano Operators
41(18)
3.1 Nakano type operators
41(4)
3.2 Bounds for eigenvalues of Nakano operators
45(4)
3.3 Dual Nakano sets
49(3)
3.4 Upper bounds for determinants of Nakano operators
52(3)
3.5 Lower bounds for determinants of Nakano operators
55(2)
3.6 Comments to
Chapter 3
57(2)
4 Determinants of Orlicz Type Operators
59(6)
4.1 The Orlicz type operators
59(2)
4.2 Upper bounds for determinants of Orlicz type operators
61(1)
4.3 Lower bounds for determinants of Orlicz type operators
62(1)
4.4 Comments to
Chapter 4
63(2)
5 Determinants of p-summing Operators
65(12)
5.1 Definitions and preliminaries
65(2)
5.2 Regularized determinants in quasi-normed ideals
67(3)
5.3 Perturbations of operators from Γp
70(2)
5.4 p-summing operators
72(1)
5.5 Hille-Tamarkin integral operators in LP
73(1)
5.6 Hille-Tamarkin infinite matrices in lp
74(1)
5.7 Comments to
Chapter 5
75(2)
6 Multiplicative Representations of Resolvents
77(18)
6.1 Representations of resolvents in a Euclidean space
77(7)
6.1.1 The first multiplicative representation for resolvents
77(5)
6.1.2 The second representation for resolvents
82(2)
6.2 Triangular representations of compact operators
84(2)
6.3 Representations for resolvents of compact operators
86(6)
6.3.1 Operators with complete systems of root vectors
86(1)
6.3.2 Multiplicative integrals
87(1)
6.3.3 The resolvent of a Volterra operator
88(2)
6.3.4 General compact operators
90(2)
6.4 Formulas for determinants and resolvents of nuclear operators
92(1)
6.5 Comments to
Chapter 6
93(2)
7 Inequalities Between Determinants and Inverse Operators
95(26)
7.1 Inequalities for finite dimensional operators
95(7)
7.1.1 The first inequality
95(1)
7.1.2 Proof of Theorem 7.1.1
96(1)
7.1.3 Auxiliary results
97(3)
7.1.4 The second inequality
100(2)
7.2 Nuclear operators
102(3)
7.3 Carleman's inequality for Hilbert-Schmidt operators
105(6)
7.4 Carleman's type inequalities for Schatten-von Neumann operators
111(6)
7.4.1 The general case
111(4)
7.4.2 Normal Schatten-von Neumann operators
115(2)
7.5 Positive invertibility of infinite matrices
117(1)
7.6 Comments to
Chapter 7
118(3)
8 Bounds for Eigenvalues and Determinants via Self-Commutators
121(14)
8.1 Series of eigenvalues
121(1)
8.2 Proof of Theorem 8.1.1
122(3)
8.3 Partial sums of eigenvalues
125(1)
8.4 Proof of Theorem 8.3.1
125(3)
8.5 Estimates for determinants via self-commutators
128(1)
8.6 Bounds for determinants via Hermitian components
128(1)
8.7 Proof of Theorem 8.6.1
129(1)
8.8 A sharp bound for the self-commutator
130(3)
8.8.1 Statement of the result
130(2)
8.8.2 Proof of Theorem 8.8.1
132(1)
8.9 Comments to
Chapter 8
133(2)
9 Spectral Variations of Compact Operators in a Hilbert Space
135(24)
9.1 Estimates for resolvents of finite dimensional operators
135(1)
9.2 Proof of Theorem 9.1.2
136(3)
9.3 Resolvents of Hilbert-Schmidt operators
139(2)
9.4 Resolvents of Schatten-von Neumann operators
141(1)
9.5 Spectral variations of operators in a Banach space
142(1)
9.6 Perturbations of finite dimensional operators
143(5)
9.6.1 Application of Lemma 9.5.1
143(2)
9.6.2 Application of the Hadamard inequality
145(3)
9.7 Spectral variations of compact operators
148(2)
9.8 An additional identity for resolvents
150(4)
9.9 Eigenvectors of perturbed operators
154(2)
9.10 Comments to
Chapter 9
156(3)
10 Discrete Spectra of Compactly Perturbed Normal Operators
159(12)
10.1 The counting function
159(2)
10.2 Proof of Theorem 10.1.1
161(3)
10.3 Eigenvalues in different domains
164(3)
10.4 Jacobi operators
167(1)
10.5 Operators in a Banach space
168(1)
10.6 Comments to
Chapter 10
169(2)
11 Perturbations of Non-Normal Noncompact Operators
171(20)
11.1 Operators with Hilbert-Schmidt components
171(2)
11.2 Proof of Theorem 11.1.1
173(5)
11.2.1 Maximal chains
173(3)
11.2.2 Diagonal and nilpotent parts
176(2)
11.3 Interpolation in the scale of Schatten-von Neumann operators
178(3)
11.4 Inequalities between components of quasi-nilpotent operators
181(3)
11.5 Operators with Schatten-von Neumann Hermitian components
184(2)
11.6 Operators close to unitary ones
186(1)
11.7 Proofs of Theorem 11.6.1
187(3)
11.7.1 The first proof
187(1)
11.7.2 The second proof
188(2)
11.8 Comments to
Chapter 11
190(1)
12 Operators on Tensor Products of Euclidean Spaces and Matrix Equations
191(18)
12.1 Preliminaries
192(3)
12.2 Simultaneously triangularizable operators
195(1)
12.3 Linear matrix equations
196(3)
12.4 Perturbations of matrix equations
199(2)
12.5 Differentiability of solutions to matrix equations with a parameter
201(2)
12.6 Bounds for determinants of bilinear operators close to triangular ones
203(2)
12.7 Perturbations of invariant subspaces of matrices
205(2)
12.8 Comments to
Chapter 12
207(2)
13 Two-Parameter Matrix Eigenvalue Problems
209(6)
13.1 Statement of the result
209(2)
13.2 Proof of Theorem 13.1.1
211(2)
13.3 A bound for the spectral radius of K0-1Kj
213(1)
13.4 Comments to
Chapter 13
214(1)
Bibliography 215(6)
List of Symbols 221(2)
Index 223
Before his retirement in 2009, Michael I. Gil was a professor at the Ben Gurion University of the Negev, Beer Sheva, Israel. He has authored more than 250 articles in scientific journals and 9 books.
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