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Full Description
This book concerns matrix and operator equations that are widely applied in various disciplines of science to formulate challenging problems and solve them in a faithful way. The main aim of this contributed book is to study several important matrix and operator equalities and equations in a systematic and self-contained fashion. Some powerful methods have been used to investigate some significant equations in functional analysis, operator theory, matrix analysis, and numerous subjects in the last decades.
The book is divided into two parts: (I) Matrix Equations and (II) Operator Equations.
In the first part, the state-of-the-art of systems of matrix equations is given and generalized inverses are used to find their solutions. The semi-tensor product of matrices is used to solve quaternion matrix equations. The contents of some chapters are related to the relationship between matrix inequalities, matrix means, numerical range, and matrix equations. In addition, quaternion algebras and their applications are employed in solving some famous matrix equations like Sylvester, Stein, and Lyapunov equations. A chapter devoted to studying Hermitian polynomial matrix equations, which frequently arise from linear-quadratic control problems. Moreover, some classical and recently discovered inequalities for matrix exponentials are reviewed.
In the second part, the latest developments in solving several equations appearing in modern operator theory are demonstrated. These are of interest to a wide audience of pure and applied mathematicians. For example, the Daugavet equation in the linear and nonlinear setting, iterative processes and Volterra-Fredholm integral equations, semicircular elements induced by connected finite graphs, free probability, singular integral operators with shifts, and operator differential equations closely related to the properties of the coefficient operators in some equations are discussed.
The chapters give acomprehensive account of their subjects. The exhibited chapters are written in a reader-friendly style and can be read independently. Each chapter contains a rich bibliography. This book is intended for use by both researchers and graduate students of mathematics, physics, and engineering.
Contents
Preface.- Part I Matrix Equations.- Chapter 1. Existence and Representations of Solutions to Some Constrained Systems of Matrix Equations.- Chapter 2. Quaternion Two-Sided Matrix Equations with Specific Constraints.- Chapter 3. Matrices over Quaternion Algebras.- Chapter 4. Direct Methods of solving quaternion matrix equation based on STP.- Chapter 5. Geometric Mean and Matrix Quadratic Equations.- Chapter 6. Yang-Baxter-like Matrix Equation: A Road Less Taken.- Chapter 7. Hermitian Polynomial Matrix Equations and Applications.- Chapter 8. Inequalities for Matrix Exponentials and Their Extensions to Lie Groups.- Chapter 9. Numerical Ranges of Operators and Matrices.- Part II Operator Equations.- Chapter 10. Stability and Controllability of Operator Differential Equations.- Chapter 11. On Singular Integral Operators with Shifts.- Chapter 12. Berezin number and norm inequalities for operators in Hilbert and semi-Hilbert spaces.- Chapter 13. Norm Equalities for Derivations.- Chapter 14. On Semicircular Elements Induced by Connected Finite Graphs.- Chapter 15. Hilbert C*-module for analyzing structured data.- Chapter 16. Iterative Processes and Integral Equations of the Second Kind.- Chapter 17. The Daugavet equation: linear and non-linear recent results.
