Perturbations of operator functions in a hilbert space

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Abstract

Let A and à 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 Ã. Let S N2 and S N1 be the ideals of Hilbert-Schmidt and nuclear operators, respectively. In the paper, a sharp estimate for the norm of f (A)-f(Ã) is established, provided A and à have the so called Hilbert-Schmidt property. In particular, A has the Hilbert-Schmidt property, if one of the following conditions holds: A - A* ∈ S N2, or AA* - I ∈ S N1. Here A* is adjoint to A, and I is the unit operator. Our results are new even in the finite dimensional case.

Original languageEnglish
Pages (from-to)108-115
Number of pages8
JournalCommunications in Mathematical Analysis
Volume13
Issue number2
StatePublished - 1 Dec 2012

Keywords

  • Non-selfadjoint operators
  • Operator functions
  • Operators "close" to unitary ones
  • Operators with Hilbert-Schmidt Hermitian components
  • Perturbations

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