On the essential norms of Toeplitz operators with continuous symbols

TL;DR

This paper investigates the essential norms of Toeplitz operators with continuous symbols on Hardy spaces, demonstrating that the norm can be strictly greater than the symbol's supremum and providing bounds related to the symbol's $L^ Infty$ norm.

Contribution

It answers a longstanding question by showing the essential norm can exceed the symbol's $L^ Infty$ norm and establishes new bounds for these norms depending on $p$.

Findings

The essential norm can be strictly greater than the $L^ Infty$ norm of the symbol.

The essential norm is at most twice the $L^ Infty$ norm of the symbol.

Derived $p$-dependent estimates for the essential norm.

Abstract

It is well known that the essential norm of a Toeplitz operator on the Hardy space $H^p(\mathbb{T})$, $1 < p < \infty$ is greater than or equal to the $L^\infty(\mathbb{T})$ norm of its symbol. In 1988, A. B\"ottcher, N. Krupnik, and B. Silbermann posed a question on whether or not the equality holds in the case of continuous symbols. We answer this question in the negative. On the other hand, we show that the essential norm of a Toeplitz operator with a continuous symbol is less than or equal to twice the $L^\infty(\mathbb{T})$ norm of the symbol and prove more precise $p$-dependent estimates.