Normal approximants to binormal operators

Abstract

Let T=[T_ij], i,j=1,2,…,m, be a block operator whose entries T_ij are commuting normal operators on a Hilbert space. We give a simple proof of the known fact that such operators can be reduced to an upper triangular form via a unitary conjugation. Our proof brings out some useful features of the triangular form. When m=2 we find the closest normal operator to the binormal operator T with respect to every unitarily invariant norm. This is a generalization of a result of J. Phillips, who solved this approximation problem for the operator bound norm.