Finite-rank perturbations of positive operators and isometries

Man-Duen Choi ; Pei Yuan Wu

Studia Mathematica, Tome 173 (2006), p. 73-79 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ≤ ∞) 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.

Détail

Publié le : 2006-01-01

Zbl 1087.47014

EUDML-ID : urn:eudml:doc:286535