Finite-rank perturbations of positive operators and isometries

Abstract

We completely characterize the ranks of A - B and A(1/2) - B-1/2 for operators A and B on a Hilbert space satisfying A >= B >= 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and m = rank(A(1/2) - B-1/2) for some operators A and B with A >= B >= 0 on a Hilbert space of dimension n (1 <= n <= infinity) if and only if l = m = 0 or 0 < l <= m <= n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that of rank(A(1/2) - B-1/2). For two isometries, we give necessary and sufficient conditions in order that they be finite-rank perturbations of each other. One such condition says that, for isometries A and B, A - B has finite rank if and only if A = (I + F)B for some unitary operator I + F with finite-rank F. Another condition is in terms of the parts in the Wold-Lebesgue decompositions of the nonunitary isometries A and B.