Nonself-adjoint operators with almost hermitian spectrum: Weak annihilators

A. V. Kiselev, S. N. Naboko

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3 Citations (SciVal)

Abstract

We consider nonself-adjoint nondissipative trace class additive perturbations L=A+iV of a bounded self-adjoint operator A in a Hilbert space H. The main goal is to study the properties of the singular spectral subspace N i 0 of L corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint nondissipative operators. To some extent, the properties of N i 0 resemble those of the singular spectral subspace of a self-adjoint operator. Namely, we prove that L and the adjoint operator L* are weakly annihilated by some scalar bounded outer analytic functions if and only if both of them satisfy the condition N i 0=H. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.

Original languageEnglish
Pages (from-to)192-201
Number of pages10
JournalFunctional Analysis and Its Applications
Volume38
Issue number3
DOIs
Publication statusPublished - 1 Jul 2004

Bibliographical note

Funding Information:
∗The authors acknowledge support received from the RFBR (grant No. 03-01-00090) and IRCSET.

Keywords

  • almost Hermitian spectrum
  • annihilator
  • functional model
  • Lagrange optimality principle
  • nonself-adjoint operator

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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