The lower order of lower triangular operators and minimal rank extensions (Q1102512)

Let Z and Y be Hilbert spaces with chains of orthogonal projections P and Q, respectively, and let L(T;P,Q) denote the lower triangular part of the operator T:Z\(\to Y\) relative to P and Q. Let T:Z\(\to Y\) be given and consider all K:Z\(\to Y\) such that \(L(K;P,Q)=L(T;P,Q).\) If among those K there is one of finite rank, there is one of minimal rank. A formula for this minimal rank (lower order) in terms of the given lower triangular part L(T;P,Q) is obtained. Also a construction for all minimal rank extensions is given. Matrices (finite, semiinfinite) and integral operators are treated as special cases. For these operators the minimal rank extension problem corresponds to a minimal realization problem in systems theory.