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Fredholmness and Compactness of Truncated Toeplitz and Hankel Operators

Integral Equations and Operator Theory volume 82, pages 451–467 (2015)Cite this article

Abstract

We prove the spectral mapping theorem \({\sigma_e(A_\phi) = \phi(\sigma_e(A_z))}\) for the Fredholm spectrum of a truncated Toeplitz operator \({A_\phi}\) with symbol \({\phi}\) in the Sarason algebra \({\mathcal{C}+H^{\infty}}\) acting on a coinvariant subspace \({K_\theta}\) of the Hardy space H 2. Our second result says that a truncated Hankel operator on the subspace \({K_\theta}\) generated by a one-component inner function \({\theta}\) is compact if and only if it has a continuous symbol. We also suppose a description of truncated Toeplitz and Hankel operators in Schatten classes \({S^{p}}\).

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Authors and Affiliations

  1. Chebyshev Laboratory, Department of Mathematics and Mechanics, St.Petersburg State University, 29b, 14th Line, 199178, St.Petersburg, Russia

    R.V. Bessonov

  2. St.Petersburg Department of Steklov Mathematical Institute, Russian Academy of Science, 27, Fontanka, 191023, St.Petersburg, Russia

    R.V. Bessonov

  3. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    R.V. Bessonov

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  1. R.V. Bessonov
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Correspondence to R.V. Bessonov.

Additional information

This work is partially supported by RFBR grants 12-01-31492, 14-01-00748, by ISF grant 94/11, by JSC “Gazprom Neft” and by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026.

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Bessonov, R. Fredholmness and Compactness of Truncated Toeplitz and Hankel Operators. Integr. Equ. Oper. Theory 82, 451–467 (2015). https://doi.org/10.1007/s00020-014-2177-2

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  • DOI: https://doi.org/10.1007/s00020-014-2177-2

Mathematics Subject Classification

  • Primary 47B35

Keywords

  • Truncated Toeplitz operators
  • Truncated Hankel operators
  • Spectral mapping theorem
  • Schatten ideal