Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$

TL;DR

This paper investigates the spectral properties of Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$, revealing that such operators are unbounded if their symbols have jump discontinuities, unlike in other Hardy spaces.

Contribution

It demonstrates that the spectral theory for piecewise continuous symbols does not extend to $H^1$, showing that Toeplitz operators with jump discontinuities are never bounded on this space.

Findings

Toeplitz operators with jump discontinuities are unbounded on $H^1$

Spectral descriptions for piecewise continuous symbols do not extend to $H^1$

Contrast with known results for $H^p$, $1<p<

Abstract

The geometric descriptions of the (essential) spectra of Toeplitz operators with piecewise continuous symbols are among the most beautiful results about Toeplitz operators on Hardy spaces $H^p$ with $1<p<\infty$. In the Hardy space $H^1$, the essential spectra of Toeplitz operators are known for continuous symbols and symbols in the Douglas algebra $C+H^\infty$. It is natural to ask whether the theory for piecewise continuous symbols can also be extended to $H^1$. We answer this question in negative and show in particular that the Toeplitz operator is never bounded on $H^1$ if its symbol has a jump discontinuity.