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Finite-rank perturbations of positive operators and isometries

100%

Choi M.-D., Wu P.

Studia Mathematica

2006

tom 173

nr 1

73-79

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.

Direct sums of irreducible operators

81%

Fang J., Jiang C.-L., Wu P.

Studia Mathematica

2003

tom 155

nr 1

37-49

It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the C*-structure of the commutant of the von Neumann algebra generated by T.

Ograniczanie wyników

2 Studia Mathematica

2 Wu P.

1 Choi M.-D.

1 Fang J.

1 Jiang C.-L.

1 2006

1 2003

Wyniki wyszukiwania

Finite-rank perturbations of positive operators and isometries

Direct sums of irreducible operators