Block Matrices of Operators

In this chapter, we introduce the concept of \(n\times n\) block matrices of operators, or simply operator matrices, which are of great importance. They allow us to represent an operator on a space using simpler operators on its specific subspaces. We provide an in-depth study of dilation theory and give several characterizations of the positivity of \(2\times 2\) operator matrices of the form \(\begin{bmatrix}A & X \\ X^* & B\end{bmatrix}\) . We also investigate the properties of \(2\times 2\) matrices with entries in a \(C^*\) -algebra. Finally, we utilize the power of operator matrices to derive a variety of inequalities related to eigenvalues and norms of matrices.