Large Essential norm of Toeplitz operators and Hankel operators on the weighted Bergman space

TL;DR

This paper investigates the essential norm of Hankel operators on weighted Bergman spaces, showing it equals the distance to compact Hankel operators and is realized by infinitely many such operators with continuous symbols.

Contribution

It extends the understanding of essential norms of Hankel operators from Hardy to weighted Bergman spaces, establishing new realization results.

Findings

Essential norm equals the distance to compact Hankel operators.

Infinitely many compact Hankel operators realize the essential norm.

Compact Hankel operators with continuous symbols vanish on the boundary.

Abstract

In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.