New spaces of matrices with operator entries

Abstract

In this paper, we will consider matrices with entries in the space of operators 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 nite 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 identied with the space of continuous functions with values in B(H). We shall also introduce matricial versions with operator entries of classical spaces of holomorphic functions such as H 1 (D) and A(D) when dealing with upper triangular matrices.

Key words: Schur product, Toeplitz matrix, Schur multiplier, vector-valued measure, vector-valued function.