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October 2019 Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$
Santeri Miihkinen, Jani Virtanen
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Ark. Mat. 57(2): 429-435 (October 2019). DOI: 10.4310/ARKIV.2019.v57.n2.a9

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 $1 \lt p \lt \infty$. 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.

Funding Statement

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.

Citation

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Santeri Miihkinen. Jani Virtanen. "Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$." Ark. Mat. 57 (2) 429 - 435, October 2019. https://doi.org/10.4310/ARKIV.2019.v57.n2.a9

Information

Received: 6 August 2018; Revised: 14 December 2018; Published: October 2019
First available in Project Euclid: 16 April 2020

zbMATH: 07114513
MathSciNet: MR4018761
Digital Object Identifier: 10.4310/ARKIV.2019.v57.n2.a9

Subjects:
Primary: 47B35
Secondary: 30H10

Rights: Copyright © 2019 Institut Mittag-Leffler

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Vol.57 • No. 2 • October 2019
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