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## Studia Mathematica

2006 | 173 | 1 | 73-79
Tytuł artykułu

### Finite-rank perturbations of positive operators and isometries

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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.
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Kategorie tematyczne
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Tom
Numer
Strony
73-79
Opis fizyczny
Daty
wydano
2006
Twórcy
autor
• Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
autor
• Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
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