On lower eigenvalue estimates for Toeplitz operators with radial symbols in Bergman spaces

TL;DR

This paper investigates eigenvalue decay rates of Toeplitz operators with radial, sign-variable symbols in Bergman spaces, revealing conditions under which eigenvalues decay faster or slower depending on symbol decay properties.

Contribution

It establishes bounds on eigenvalue decay for Toeplitz operators with variable sign radial symbols, extending understanding of spectral behavior in Bergman spaces.

Findings

Eigenvalues cannot decay too fast if the symbol has compact support or decays rapidly.

Eigenvalues may decay faster than for the absolute value of the symbol if decay is insufficient.

Eigenvalue decay rates depend critically on the decay properties of the radial symbol.

Abstract

We consider Toeplitz operators in different Bergman type spaces, having radial symbols with variable sign. We show that if the symbol has compact support or decays rapidly, the eigenvalues of such operators cannot decay too fast, essentially faster than for a sign-definite symbol with the same kind. On the other hand, if the symbol decays not sufficiently rapidly, the eigenvalues of the corresponding operator may decay faster than for the operator corresponding to the absolute value of the symbol.