Fixed points of the Berezin transform on Fock-type 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 $m.$

arXiv commentMinor changes. To appear in Proc. Amer. Math. Soc