Small Hankel operators on generalized Fock spaces

TL;DR

This paper studies small Hankel operators on generalized Fock spaces defined by exponential weights, explicitly computes their kernels, and characterizes their boundedness, compactness, and Hilbert-Schmidt properties.

Contribution

It provides explicit formulas for Bergman kernels and new characterizations of operator boundedness, compactness, and Hilbert-Schmidt criteria on these generalized Fock spaces.

Findings

Explicit Bergman kernel formulas for $F^{2, ext{ell}}_{ ext{alpha}}$

Characterization of boundedness and compactness of small Hankel operators

Criteria for when these operators are Hilbert-Schmidt

Abstract

We consider Fock spaces $F^{p,\ell}_{\alpha}$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-\alpha |z|^{2\ell}}$, where $\alpha>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_{\alpha}$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,\alpha}$ on $F^{p,\ell}_{\alpha}$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,\alpha}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_{\alpha}$.