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

Santeri Miihkinen; Jani A. Virtanen

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

Functional Analysis 2019-01-09 v2 Complex Variables Spectral Theory

Authors: Santeri Miihkinen , Jani A. Virtanen

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.

Keywords

toeplitz operators piecewise continuous symbols hardy space h1 essential spectrum jump discontinuity bounded operators

Cite

@article{arxiv.1808.01788,
  title  = {Toeplitz operators with piecewise continuous symbols on the Hardy space $H^1$},
  author = {Santeri Miihkinen and Jani A. Virtanen},
  journal= {arXiv preprint arXiv:1808.01788},
  year   = {2019}
}

Comments

To appear in Ark. Mat.. 6 pages, 1 figure. Minor clarifications to the statements of Theorem 6 and Corollary 8

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

Abstract

Keywords

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