Fixed points of the Berezin transform on Fock-type spaces

TL;DR

This paper investigates the fixed points of the Berezin transform on Fock-type spaces with exponential weights, revealing that polynomial fixed points are harmonic except for certain parameters, with some cases showing harmonicity universally.

Contribution

It characterizes polynomial fixed points of the Berezin transform on Fock-type spaces, showing they are harmonic for most parameters and identifying special cases where this holds universally.

Findings

Polynomial fixed points are harmonic for most weights.

Certain parameter values allow all fixed points to be harmonic.

The study extends understanding of Berezin transform fixed points on weighted spaces.

Abstract

We study the fixed points of the Berezin transform on the Fock-type spaces $F_m^2$ with the weight $e^{-|z|^m}, m > 0.$ It is known that the Berezin transform is well-defined on the polynomials in $z$ and $\overline{z}$. In this paper we focus on the polynomial fixed points and we show that these polynomials must be harmonic, except possibly for countably many $m \in (0, \infty).$ We also show that, in some particular cases, the fixed point polynomials are harmonic for all $m.$