Hankel operators on the Fock-Sobolev spaces

TL;DR

This paper investigates the properties of Hankel operators on Fock-Sobolev spaces, focusing on boundedness and compactness, and relates these to BMO and VMO spaces, extending classical characterizations from Bergman and weighted Fock spaces.

Contribution

It provides new operator-theoretic characterizations of Hankel operators on Fock-Sobolev spaces, extending classical results to this broader context.

Findings

Characterization of bounded Hankel operators via BMO spaces.

Characterization of compact Hankel operators via VMO spaces.

Analysis of Berezin transform of Hankel operators on Fock-Sobolev spaces.

Abstract

In this paper, we study operator-theoretic properties (boundedness and compactness) of Hankel operators on the Fock-sobolev spaces $ \mathscr{F}^{p,m} $ in terms of $ \mathcal{BMO}_r^p $ and $ \mathcal{VMO}_r^p $ spaces, respectively, for a non-negative integers $ m $, $ 1 \leq p < \infty $ and $ r > 0 $. Along the way, we also study Berezin transform of Hankel operators on $ \mathscr{F}^{p,m} $. The results in this article are analogous to Zhu's characterization and Per\"al\"a's characterization of bounded and compact Hankel operators on the Bergman spaces of unit disc and the weighted Fock spaces, respectively.