This book presents a wide panorama of methods to investigate the spectral properties of block operator matrices. Particular emphasis is placed on classes of block operator matrices to which standard operator theoretical methods do not readily apply: non-self-adjoint block operator matrices, block operator matrices with unbounded entries, non-semibounded block operator matrices, and classes of block operator matrices arising in mathematical physics.
The main topics include: localization of the spectrum by means of new concepts of numerical range; investigation of the essential spectrum; variational principles and eigenvalue estimates; block diagonalization and invariant subspaces; solutions of algebraic Riccati equations; applications to spectral problems from magnetohydrodynamics, fluid mechanics, and quantum mechanics.
Contents: Bounded Block Operator Matrices: ; The Quadratic Numerical Range; Special Classes of Block Operator Matrices; Spectral Inclusion; Estimates of the Resolvent; Corners of the Quadratic Numerical Range; Schur Complements and Their Factorization; Block Diagonalization; Spectral Supporting Subspaces; Variational Principles for Eigenvalues in Gaps; J-Self-Adjoint Block Operator Matrices; The Block Numerical Range; Numerical Ranges of Operator Polynomials; Gershgorin's Theorem for Block Operator Matrices; Unbounded Block Operator Matrices: ; Relative Boundedness and Relative Compactness; Closedness and Closability of Block Operator Matrices; Spectrum and Resolvent; The Essential Spectrum; Spectral Inclusion; Symmetric and J-Symmetric Block Operator Matrices; Dichotomous Block Operator Matrices and Riccati Equations; Block Diagonalization and Half Range Completeness; Uniqueness Results for Solutions of Riccati Equations; Variational Principles; Eigenvalue Estimates; Applications in Mathematical Physics: ; Upper Dominant Block Operator Matrices in Magnetohydrodynamics; Diagonally Dominant Block Operator Matrices in Fluid Mechanics; Off-Diagonally Dominant Block Operator Matrices in Quantum Mechanics.
Author(s): Christiane Tretter
Publisher: Imperial College Press
Year: 2008
Language: English
Pages: 297
Contents......Page 32
Preface......Page 6
Introduction......Page 8
1.1 The quadratic numerical range......Page 34
1.2 Special classes of block operator matrices......Page 44
1.3 Spectral inclusion......Page 51
1.4 Estimates of the resolvent......Page 59
1.5 Corners of the quadratic numerical range......Page 62
1.6 Schur complements and their factorization......Page 68
1.7 Block diagonalization......Page 75
1.8 Spectral supporting subspaces......Page 80
1.9 Variational principles for eigenvalues in gaps......Page 92
1.10 J -self-adjoint block operator matrices......Page 95
1.11 The block numerical range......Page 103
1.12 Numerical ranges of operator polynomials......Page 115
1.13 Gershgorin's theorem for block operator matrices......Page 119
2.1 Relative boundedness and relative compactness......Page 124
2.2 Closedness and closability of block operator matrices......Page 132
2.3 Spectrum and resolvent......Page 144
2.4 The essential spectrum......Page 149
2.5 Spectral inclusion......Page 162
2.6 Symmetric and J -symmetric block operator matrices......Page 175
2.7 Dichotomous block operator matrices and Riccati equations......Page 187
2.7.1 Essentially self-adjoint block operator matrices......Page 190
2.7.2 Non-self-adjoint block operator matrices......Page 198
2.8 Block diagonalization and half range completeness......Page 207
2.9 Uniqueness results for solutions of Riccati equations......Page 213
2.10 Variational principles......Page 226
2.11 Eigenvalue estimates......Page 238
3.1 Upper dominant block operator matrices in magnetohydrodynamics......Page 250
3.2 Diagonally dominant block operator matrices in uid mechanics......Page 255
3.3.1 Dirac operators in R3......Page 260
3.3.2 The angular part of the Dirac equation in the Kerr-Newman metric......Page 267
Bibliography......Page 272
Index......Page 294
