Numerical Ranges of Hilbert Space Operators (Encyclopedia of Mathematics and its Applications)

Starting with elementary operator theory and matrix analysis, this book introduces the basic properties of the numerical range and gradually builds up the whole numerical range theory. Over 400 assorted problems, ranging from routine exercises to published research results, give you the chance to put the theory into practice and test your understanding. Interspersed throughout the text are numerous comments and references, allowing you to discover related developments and to pursue areas of interest in the literature. Also included is an appendix on basic convexity properties on the Euclidean space. Targeted at graduate students as well as researchers interested in functional analysis, this book provides a comprehensive coverage of classic and recent works on the numerical range theory. It serves as an accessible entry point into this lively and exciting research area. Contents Preface List of Symbols Introduction: Preliminaries in Operator Theory I.1 Basic Properties I.2 Spectral Theory I.3 Special Types of Operators I.4 Matrix Theory I.5 C∗-Algebra Theory I.7 Compression and Dilation 1 Numerical Range 1.1 Basic Properties 1.2 Relation to Spectrum 1.3 Boundary and Interior 1.4 Limit and Transforms 1.5 Parameters Problems 2 Numerical Ranges of Special Operators 2.1 Quadratic Operator 2.2 Normal Operator 2.3 Hyponormal and Toeplitz Operators 2.4 Weighted Shift 2.5 Composition Operator 2.6 Attainment Problem Problems 3 Numerical Contraction 3.1 Numerical Radius 3.2 Numerical Contraction 3.3 Power Inequality and Ando’s Theorem 3.4 Commuting Product Problems 4 Algebraic and Essential Numerical Ranges 4.1 Algebraic Numerical Range 4.2 Essential Numerical Range 4.3 Commutator and Zero-Diagonal Operator 4.4 Total Dilation 4.5 Compact Operator Problems 5 Numerical Range and Dilation 5.1 Unitary Dilation 5.2 Berger Power Dilation 5.3 Nilpotent Dilation 5.4 Spectral Set Problems 6 Numerical Range of Finite Matrix 6.1 Kippenhahn Curve 6.2 3-by-3 Matrix 6.3 Anderson’s Theorem 6.4 Line Segment 6.5 Nonnegative Matrix Problems 7 Numerical Range of S_n-Matrix 7.1 Basic Properties 7.2 Poncelet’s Porism 7.3 Generalized Brianchon-Ceva and Lucas-Siebeck Theorems 7.4 Extension to Inflation of S(φ) 7.5 Norms of Powers and Gau–Wu Number Problems 8 Generalized Numerical Ranges 8.1 Joint Numerical Range 8.2 C-Numerical Range 8.3 q-Numerical Range and Davis–Wielandt Shell 8.4 Matricial Ranges 8.5 Higher-Rank Numerical Range 8.6 Zero-Dilation Index Problems Appendix. Convex Set A.1 Basic Properties A.2 Boundary Parametrization References [19] [45] [70] [97] [122] [147] [171] [197] [221] [246] [271] [298] [322] [348] [374] [399] [424] [451] [476] [504] [529] [555] [581] Index ABC DEFGHIJK LMNOPQR STUVWZ "Starting with elementary operator theory and matrix analysis, this book introduces the basic properties of the numerical range and gradually builds up the whole numerical range theory. More than 400 assorted problems, ranging from routine exercises to published research results, give you the chance to put the theory into practice and test your understanding. Interspersed throughout the text are numerous comments and references, allowing you to discover related developments and to pursue areas of interest in the literature. Also included is an appendix on basic convexity properties on the Euclidean space. Targeted at graduate students as well as researchers interested in functional analysis, this book provides a comprehensive coverage of classic and recent works on the numerical range theory. It serves as an accessible entry point into this lively and exciting research area"-- Provided by publisher