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This book is a self-contained advanced monograph on inequalities involving the numerical radius of bounded linear operators acting on complex Hilbert spaces. The study of numerical range and numerical radius has a long and distinguished history starting from the Rayleigh quotients used in the 19th century to nowadays applications in quantum information theory and quantum computing.
This monograph is intended for use by both researchers and graduate students of mathematics, physics, and engineering who have a basic background in functional analysis and operator theory. The book provides several challenging problems and detailed arguments for the majority of the results. Each chapter ends with some notes about historical views or further extensions of the topics. It contains a bibliography of about 180 items, so it can be used as a reference book including many classical and modern numerical radius inequalities.
Preliminaries
General Notation and Terminology
Elementary Functional Analysis
Functional Calculus for Selfadjoint Operators
Spectral Representation of Selfadjoint Operators
Block Matrices of Operators
Fundamental Numerical Radius Inequalities
Basic Properties of Numerical Range
Inequalities Concerning One Operator
Inequalities Related to Davis–Wielandt Radius
Inequalities Involving More Than One Operator
More Applications of Cartesian Decomposition
Bounds of the Numerical Radius Using Buzano's Inequality
Upper Bounds of the Numerical Radius
Generalized Buzano's Inequality
Vector Inequalities for Projections
p-Numerical Radius Inequalities of an n-Tuple of Operators
Basic Properties of p-Numerical Radius
Inequalities of wp for Pairs of Operators
Inequalities of wp for n-Tuples of Operators
Hypo-p-Numerical Radius
Reverse Inequalities for Hypo-Euclidean Numerical Radius
Numerical Radius Inequalities of Product of Operators
Inequalities Involving Adjoint Operators
Inequalities for Product of Two or More Operators
Numerical Radius of Operator Matrices and Applications
Basic Inequalities of times Operator Matrices
Inequalities of times Off-Diagonal Operator Matrices
Bounds for the Zeros of a Polynomial
Bounded Region Containing All the Zeros of a Polynomial
Operator Space Numerical Radius of times Block Matrices
Complete Numerical Radius Norm
Inequalities for Operator Space Numerical Radius
A-Numerical Radius Inequalities in Semi-Hilbertian Spaces
Basic Definitions and Notations
Fundamental Properties
Inequalities Concerning One Operator
Inequalities Concerning Two Operators
Inequalities Concerning Block Matrices
Research Problems
Appendix Bibliography