A Class of Schur Multipliers of Matrices with Operator Entries

In this paper, we will consider matrices with entries in the space of operators B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(H)$$\end{document}, 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.