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Fredholmness and Compactness of Truncated Toeplitz and Hankel Operators

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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Correspondence to R.V. Bessonov.

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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