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Essential positivity

A. Perälä, J. A. Virtanen

Abstract

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the nonnegative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space.

Essential positivity

Abstract

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the nonnegative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space.
Paper Structure (4 sections, 9 theorems, 45 equations)

This paper contains 4 sections, 9 theorems, 45 equations.

Table of Contents

  1. Essentially positive operators on Hilbert spaces
  2. The Berezin transform
  3. Toeplitz operators on the Hardy space
  4. Toeplitz operators on the Bergman space

Key Result

Proposition 2

Let $H$ be a Hilbert space and $T\in L(H)$ be self-adjoint. Then $T$ is essentially positive if and only if $T+K(H)$ is positive in $L(H)/K(H)$.

Theorems & Definitions (19)

  • Definition 1
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • Proposition 6
  • proof
  • ...and 9 more