Integral Equations and Operator Theory

, Volume 82, Issue 4, pp 451–467 | Cite as

Fredholmness and Compactness of Truncated Toeplitz and Hankel Operators

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 H2. 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 \({S^{p}}\).

Mathematics Subject Classification

Primary 47B35 

Keywords

Truncated Toeplitz operators Truncated Hankel operators Spectral mapping theorem Schatten ideal 

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© Springer Basel 2014

Authors and Affiliations

  1. 1.Chebyshev Laboratory, Department of Mathematics and MechanicsSt.Petersburg State UniversitySt.PetersburgRussia
  2. 2.St.Petersburg Department of Steklov Mathematical InstituteRussian Academy of ScienceSt.PetersburgRussia
  3. 3.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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