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This book introduces several powerful techniques and fundamental ideas involving block matrices of operators, as well as matrices with elements in a C*-algebra. These techniques allow for the solution of problems that may be difficult to treat. Specifically, 2×2 operator matrices yield significant mathematical inequalities in various fields of operator theory and matrix analysis. The authors employ block matrices to simplify complicated problems. Operator matrices have garnered attention for their applications in quantum information and computing theories.
Each chapter concludes with a diverse set of exercises and problems for readers, along with references to relevant literature. Some problems pose open questions, while others challenge readers and provide suggestions for future research. This book is suitable for an advanced undergraduate or graduate course and can be used in the classroom. It also serves as a valuable resource for researchers and students in mathematics and physics who have a basic understanding of linear algebra, functional analysis, and operator theory.
Preface
Matrices and Hilbert Space Operators
Fundamental Information
Some Decompositions of Operators
Unitarily Invariant Norms
Moore–Penrose Inverse
Differences Between Real and Complex Hilbert Spaces
Exercises and Problems
Block Matrices of Operators
Operator Matrices of Size n times nn timesn
Dilation
2times2 Matrices with Entries in a upper C Superscript asteriskC*-Algebra
Positivity of 2times2 Matrices of Operators
Diagonal Blocks and Unitary Orbits
Inequalities Follow from the Positivity of 2times2 Block Matrices
Inequalities Related to Unitarily Invariant Norms and Numerical Radius
Some Inequalities Involving Eigenvalues and Singular Values
Exercises and Problems
Operator Monotone Functions and Positive Maps
Operator Monotone and Operator Convex Functions
Operator Means
Positive and Completely Positive Linear Maps
Weakly 2x2-Positive Maps
Positive Linear Maps on Matrix Algebras
Exercises and Problems
Operator Variance and Covariance
Bounds for the Operator Norm of the Covariance
Bounds for Unitarily Invariant Norms of the Covariance
Concluding Remarks
Exercises and Problems
Nonlinear Positive Maps
Lieb Maps and Their Fundamental Properties
3x3-Positivity of Nonlinear Maps
Continuity of 3x3-Positive Maps
Exercises and Problems
Bibliography
Index