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.

M. S. Moslehian is a professor of Mathematics at Ferdowsi University of Mashhad, a member of the Academy of Sciences of Iran, a TWAS fellow, and the president of the Iran. Math. Soc. His research focuses on functional analysis, operator theory, and matrix analysis. He has served as a senior associate at ICTP (Italy) and as a visiting professor at various universities in England, Sweden, and Japan. He is also the editor-in-chief of the journals "Banach J. Math. Anal.", "Ann. Funct. Anal.", and "Adv. Oper. Theory" published by Birkhäuser/Springer.

Hiroyuki Osaka is a professor in the Department of Mathematical Sciences at Ritsumeikan University, Japan. His research concerns operator algebras, operator theory, and quantum information theory. He was a postdoctoral researcher at Fields Institute (Canada), an assistant research professor at Copenhagen University (Denmark), an associate professor at Ryukyu University (Japan), and a visiting professor at several universities in the USA, India, and Poland. He is an editor of "Adv. Oper. Theory" published by Birkhäuser/Springer and "Sci. Math. Jpn." published by Inst. Soc. Math. Sci.

Preface.- 1 Matrices and Hilbert Space Operators.- 2 Block Matrices of Operators.- 3 Operator Monotone Functions and Positive Maps.- 4 Operator Variance and Covariance.- 5 Nonlinear Positive Maps.- Bibliography.