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Finite-rank perturbations of positive operators and isometries

Man-Duen ChoiPei Yuan Wu — 2006

Studia Mathematica

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 = r a n k ( 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 r a n k ( A 1 / 2 - B 1 / 2 ) . For...

Direct sums of irreducible operators

Jun Shen FangChun-Lan JiangPei Yuan Wu — 2003

Studia Mathematica

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...

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