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

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 $1 \lt p \lt \infty$. 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 the negative and show in particular that the Toeplitz operator is never bounded on $H^1$ if its symbol has a jump discontinuity.

Keywords

Toeplitz operators, Hardy spaces, Fredholm properties, essential spectrum, piecewise continuous symbols

2010 Mathematics Subject Classification

Primary 47B35. Secondary 30H10.

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S. Miihkinen was supported by the Academy of Finland project 296718. J. Virtanen was supported in part by Engineering and Physical Sciences Research Council grant EP/M024784/1.

Received 6 August 2018

Received revised 14 December 2018

Accepted 28 December 2018

Published 7 October 2019