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This book is a comprehensive and advanced exploration of trace inequalities in the context of matrices and operators acting on Hilbert spaces. Its goal is to present elegant inequalities with innovative proofs. Instead of presenting generalized versions that can be complicated and lack clarity, the book focuses on beautiful and original inequalities. Divided into eight chapters, this book is designed for researchers and graduate students in mathematics, physics, and engineering. It provides detailed explanations for most of the results and includes a variety of exercises and problems to help readers understand the content and inspire further research into advanced topics. Preface Reference Fundamentals of Matrices and Operators Operators on Hilbert Spaces Functional Calculus for Self-adjoint Operators Spectral Representation for Self-adjoint Operators Löwner Order Borel Functional Calculus Operator Matrices Tensor Product and Hadamard Product of Matrices Exercises and Problems Notes, Hints, and References References Unitarily Invariant Norms and Inequalities Trace of a Matrix Majorization Unitarily Invariant Norms Basic Inequalities Related to Unitarily Invariant Norms Exercises and Problems Notes, Hints, and References References Trace Inequalities for Positive Semidefinite Matrices Partial Solutions to the First Main Question Lieb–Thirring Trace Inequalities and the Second Main Question Ando–Hiai–Okubo Trace Inequalities Some Classical Trace Inequalities Further Applications Exercises and Problems Notes, Hints, and References References Norm Inequalities for Positive Semidefinite Matrices A Complete Solution to the First Main Question Partial Solutions to the Second Main Question Related Trace Inequalities Related Majorization Relations Inequalities Involving the Spectral Radius Inequalities Related to Arithmetic–Geometric Mean Inequalities Exercises and Problems Notes, Hints, and References References Positive Maps and Operator Means Operator Convex and Operator Monotone Functions Positive Linear Maps Operator Means Exercises and Problems Notes, Hints, and References References Golden–Thompson Trace Inequalities Golden–Thompson Trace Inequalities Strengthened Variant of the Golden–Thompson Trace Inequality Complementary Golden–Thompson Trace Inequality Norm Inequalities for Quasi alphaα-Geometric Means Reverse Inequality to Golden–Thompson Inequality Applications I (Hadamard Product Version) Applications II (Multivariable Version) Exercises and Problems Notes, Hints, and References References Quantum Relative Entropy Quantum Entropy and Quantum Relative Entropy Another Quantum Tsallis Relative Entropies Comparison of Two Quantum Tsallis Relative Entropies Exercises and Problems Notes, Hints, and References References Trace Inequalities for von Neumann Algebras Traces on Von Neumann Algebra Trace Inequalities for Quantum Statistical Mechanics Inequalities for Real Functions and the Trace Weak Majorization Inequalities for Determinants and Characterization of the Trace New Inequalities for Determinants and the Trace The Inequalities for Determinants that Characterize the Trace Noncommutative Probability Spaces Bennett Inequality Hoeffding Inequality Azuma Inequality Concluding Remarks Exercises and Problems Notes, Hints, and References References Index