A class of Schur multipliers of matrices with operator entries

TL;DR

This paper studies Schur multipliers on matrices with operator entries, focusing on Toeplitz and upper triangular matrices, and explores their connections with vector-valued function spaces.

Contribution

It introduces a class of Schur multipliers approximable by finite-diagonal matrices and analyzes their properties in the context of operator-valued matrices.

Findings

Characterization of Schur multipliers approximable by finite-diagonal matrices.

Connections established between Toeplitz matrices and vector-valued function spaces.

Insights into the structure of Schur multipliers for operator-valued matrices.

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 (left or right) Schur multipliers that can be approached in the multiplier norm by matrices with a finite number of diagonals. We will concentrate on the case of Toeplitz matrices and of upper triangular matrices to get some connections with spaces of vector-valued functions.