Essential positivity

TL;DR

This paper introduces the concept of essentially positive operators on Hilbert space, characterizes their properties, and illustrates their application in Hardy and Bergman spaces using Toeplitz operators and the Berezin transform.

Contribution

It defines essential positivity for operators, explores its fundamental properties, and demonstrates its relevance in function spaces via Toeplitz operators and the Berezin transform.

Findings

Essential spectra of these operators are contained in nonnegative reals.

Basic properties of essentially positive operators are established.

Applications in Hardy and Bergman spaces are demonstrated.

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.