This self-contained monograph unifies theorems, applications and problem solving techniques of matrix inequalities. In addition to the frequent use of methods from Functional Analysis, Operator Theory, Global Analysis, Linear Algebra, Approximations Theory, Difference and Functional Equations and more, the reader will also appreciate techniques of classical analysis and algebraic arguments, as well as combinatorial methods. Subjects such as operator Young inequalities, operator inequalities for positive linear maps, operator inequalities involving operator monotone functions, norm inequalities, inequalities for sector matrices are investigated thoroughly throughout this book which provides an account of a broad collection of classic and recent developments. Detailed proofs for all the main theorems and relevant technical lemmas are presented, therefore interested graduate and advanced undergraduate students will find the book particularly accessible. In addition to several areas of theoretical mathematics, Matrix Analysis is applicable to a broad spectrum of disciplines including operations research, mathematical physics, statistics, economics, and engineering disciplines. It is hoped that graduate students as well as researchers in mathematics, engineering, physics, economics and other interdisciplinary areas will find the combination of current and classical results and operator inequalities presented within this monograph particularly useful.
Author(s): Mohammad Bagher Ghaemi; Nahid Gharakhanlu; Themistocles M. Rassias; Reza Saadati
Series: Springer Optimization and Its Applications, 176
Publisher: Springer
Year: 2021
Language: English
Pages: 272
City: Cham
Preface
Contents
Acronyms
1 Elementary Linear Algebra Review
1.1 Operators and Matrices in Hilbert Space
2 Interpolating the Arithmetic-Geometric Mean Inequality and Its Operator Version
2.1 Refinements of the Scalar Young and Heinz Inequalities
2.2 Operator Inequalities Involving Improved Young Inequality
2.3 Advanced Refinements of the Scalar Reverse Young Inequalities
2.4 Improvements of the Operator Reverse Young Inequality
3 Operator Inequalities for Positive Linear Maps
3.1 On an Operator Kantorovich Inequality for Positive Linear Maps
3.2 A Schwarz Inequality for Positive Linear Maps
3.3 Squaring the Reverse Arithmetic-Geometric Mean Inequality
3.4 Reverses of Ando's Inequality for Positive Linear Maps
3.5 Squaring the Reverse Ando's Operator Inequality
4 Operator Inequalities Involving Operator Monotone Functions
4.1 Young Inequalities Involving Operator Monotone Functions
4.2 Eigenvalue Inequalities Involving Operator Concave Functions
4.3 Operator Aczél Inequality Involving Operator Monotone Functions
4.4 Norm Inequalities Involving Operator Monotone Functions
5 Inequalities for Sector Matrices
5.1 Haynsworth and Hartfiel Type Determinantal Inequality
5.2 Inequalities with Determinants of Perturbed Positive Matrices
5.3 Analogue of Fischer's Inequality for Sector Matrices
5.4 Analogues of Hadamard and Minkowski Inequality for Sector Matrices
5.5 Generalizations of the Brunn Minkowski Inequality
5.6 A Lewent Type Determinantal Inequality
5.7 Principal Powers of Matrices with Positive Definite Real Part
5.8 Geometric Mean of Accretive Operators
5.9 Weighted Geometric Mean of Accretive Operators and Its Applications
5.10 Ficher Type Determinantal Inequalities for Accretive-Dissipative Matrices
5.11 Extensions of Fischer's Inequality for Sector Matrices
5.12 Singular Value Inequalities of Sector Matrices
5.13 Extension of Rotfel'd Inequality for Sector Matrices
5.14 A Further Extension of Rotfel'd Inequality for Accretive-Dissipative Matrices
5.15 Hilbert-Schmidt Norm Inequalities for Accretive-Dissipative Operators
5.16 Schatten p-Norm Inequalities for Accretive-Dissipative Matrices
5.17 Schatten p-Norm Inequalities for Sector Matrices
5.18 Schatten p-Norms and Determinantal Inequalities Involving Partial Traces
5.19 Ando-Choi Type Inequalities for Sector Matrices
5.20 Geometric Mean Inequality for Sector Matrices
5.21 Weighted Geometric Mean Inequality for Sector Matrices
6 Positive Partial Transpose Matrix Inequalities
6.1 Singular Value Inequalities Related to PPT Matrices
6.2 Matrix Inequalities and Completely PPT Maps
6.3 Hiroshima's Type Inequalities for Positive Semidefinite Block Matrices
6.4 Geometric Mean and Norm Schwarz Inequality
6.5 Inequalities Involving the Off-Diagonal Block of a PPT Matrix
6.6 Unitarily Invariant Norm Inequalities of PPT Matrices
6.7 On Symmetric Norm Inequalities for Positive Block Matrices
6.8 Matrix Norm Inequalities and Majorization Relation for Singular Values
Appendix References
Index
