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.

Spectral Theory of Block Operator Matrices and Applications PDF

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