Finite-rank perturbations of positive operators and isometries

• Autorzy: Man-Duen Choi , Pei Yuan Wu
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)
• 47A55: Perturbation theory
• 47B20: Subnormal operators, hyponormal operators, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 173
• Numer: 1
• Strony: 73-79

Daty

wydano

2006

Twórcy

autor

Man-Duen Choi

• Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

autor

Pei Yuan Wu

• Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

Identyfikatory

DOI

10.4064/sm173-1-5