Unique minimal rank extensions of triangular operators

Abstract

The problem to find the minimal rank extensions of the lower triangular part of a finite rank matrix kernel appears in a natural way (see [3]) in the study of minimal representations of semi-separable integral operators or, equivalently, of minimal realizations of boundary value systems. In particular, one is interested in those kernels for which the minimal rank extension is unique. In this paper we give a full description of this class of so-called unique matrix kernels. We also define and characterize lower uniqueness for finite and semi-infinite block operator matrices. To state one of our main results, let k( t, s) be an m x n matrix kernel defined on the square [a, b] x [a, b]. Following [3] we say that k is lower separable if the lower triangular part k, of k, which is defined by kL,(t, 8) = k(t, s), a<s<t<b, admits a finite rank extension, i.e., there exists a finite rank matrix kernel h on [a, b] x [a, b] such that k, is the lower triangular part of h. Recall that for an m x n matrix kernel h on [a, fl] x [y, S] the rank of