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On non-negative operators in Krein spaces and their perturbations

Jussi Behrndt, Friedrich M. Philipp, Carsten Trunk

Abstract

One of the most important contributions of Heinz Langer in the area of operator theory in Krein spaces is the introduction of the notion of definitizable operators and the construction of the corresponding spectral function. In this note we obtain a new characterization for the subclass of non-negative operators in Krein spaces which is based on local sign type properties of the spectrum and growth conditions on the resolvent. Based on these local properties, a notion of local non-negativity for self-adjoint operators in Krein spaces is defined and it is shown that such classes of operators appear naturally as perturbations of non-negative operators.

On non-negative operators in Krein spaces and their perturbations

Abstract

One of the most important contributions of Heinz Langer in the area of operator theory in Krein spaces is the introduction of the notion of definitizable operators and the construction of the corresponding spectral function. In this note we obtain a new characterization for the subclass of non-negative operators in Krein spaces which is based on local sign type properties of the spectrum and growth conditions on the resolvent. Based on these local properties, a notion of local non-negativity for self-adjoint operators in Krein spaces is defined and it is shown that such classes of operators appear naturally as perturbations of non-negative operators.

Paper Structure

This paper contains 6 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on self-adjoint operators in Krein spaces
  3. Spectral characterization of non-negative operators
  4. Spectral characterization of locally non-negative operators
  5. Perturbations of non-negative operators
  6. Author Affiliations

Key Result

Theorem 2.5

Let $A$ be a self-adjoint operator in the Krein space $(\mathcal{H},[\cdot\,,\cdot])$, assume that $A$ is definitizable over $\Omega$, and let $\Delta\mapsto E(\Delta)$ be the local spectral function on $\Omega\cap\overline{\mathbb R}$. Then the spectral projection $E(\Delta)$ is a bounded $[\cdot\,

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 15 more