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Perturbations of Operator Functions in a Hilbert Space

Commun. Math. Anal.
Volume 13, Number 2 (2012), 108 - 115

Perturbations of Operator Functions in a Hilbert Space

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Abstract

Let A and A~ be linear bounded operators in a separable Hilbert space, and f be a function analytic on the closed convex hull of the spectra of A and A~. Let SN2 and SN1 be the ideals of Hilbert-Schmidt and nuclear operators, respectively. In the paper, a sharp estimate for the norm of f(A)−f(A~) is established, provided A and A~ have the so called Hilbert-Schmidt property. In particular, A has the Hilbert-Schmidt property, if one of the following conditions holds: AA∗∈SN2, or AA∗−ISN1. Here A∗ is adjoint to A, and I is the unit operator. Our results are new even in the finite dimensional case.