  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.
