Opuscula Math. 44, no. 3 (2024), 341-357

Opuscula Mathematica

Shifted model spaces and their orthogonal decompositions

M. Cristina Câmara
Kamila Kliś-Garlicka
Marek Ptak

Abstract. We generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift \(S^*\). In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator \(T\), with respect to another operator, follow from certain commutation relations between the two operators involved.

Keywords: model space, Toeplitz operator, Toeplitz kernel, truncated Toeplitz operator, nearly invariant, shift invariant.

Mathematics Subject Classification: 47B32, 47B35, 30H10.

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  • M. Cristina Câmara
  • ORCID iD https://orcid.org/0000-0001-9015-3980
  • Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
  • Kamila Kliś-Garlicka (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-7104-1729
  • Department of Applied Mathematics, University of Agriculture, ul. Balicka 253c, 30-198 Kraków, Poland
  • Communicated by P.A. Cojuhari.
  • Received: 2023-03-01.
  • Accepted: 2023-10-10.
  • Published online: 2024-02-15.
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Cite this article as:
M. Cristina Câmara, Kamila Kliś-Garlicka, Marek Ptak, Shifted model spaces and their orthogonal decompositions, Opuscula Math. 44, no. 3 (2024), 341-357, https://doi.org/10.7494/OpMath.2024.44.3.341

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