New spaces of matrices with operator entries

TL;DR

This paper explores matrices with operator entries on a Hilbert space, characterizing classes approximable by finite diagonals using Schur products and linking Toeplitz matrices to continuous operator-valued functions.

Contribution

It introduces a new class of operator-valued matrices approximable by finite diagonals and connects Toeplitz matrices with continuous operator-valued functions, extending classical function spaces.

Findings

Characterization of matrices approximable by finite diagonals using Schur products.

Identification of Toeplitz matrices with continuous functions valued in (H).

Extension of classical holomorphic function spaces to matrices with operator entries.

Abstract

In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in $\mathcal B(H)$. We shall also introduce matriceal versions with operator entries of classical spaces of holomorphic functions such as $H^\infty(\mathbb{D})$ and $A(\mathbb{D})$ when dealing with upper triangular matrices.