Arkiv för Matematik

Volume 57 (2019)

Number 2

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

Pages: 429 – 435



Santeri Miihkinen (Department of Mathematics, Åbo Akademi University, Turku, Finland)

Jani Virtanen (Department of Mathematics and Statistics, University of Reading, England)


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.


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

2010 Mathematics Subject Classification

Primary 47B35. Secondary 30H10.

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