Numerical range of Toeplitz and Composition operators on weighted Bergman spaces

TL;DR

This paper fully characterizes the numerical range of Toeplitz and weighted composition operators on weighted Bergman spaces with harmonic symbols, including geometric properties and the inclusion of zero.

Contribution

It provides a complete description of the numerical range for these operators and classifies certain sets as their numerical ranges, advancing understanding of their spectral properties.

Findings

Numerical range of Toeplitz operators with harmonic symbols is fully described.

The numerical range of weighted composition operators is characterized and classified.

The inclusion of zero and geometric bounds like circle and ellipse radii are computed.

Abstract

In this paper we completely describe the numerical range of Toeplitz operators on weighted Bergman spaces with harmonic symbol. We also characterize the numerical range of weighted composition operators on weighted Bergman spaces and classify some sets which are the numerical range of composition operators. We investigate the inclusion of zero in the numerical range, and compute the radius of circle and ellipse contained in the numerical range of weighted composition operators on weighted Bergman spaces.