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Riemam Surface Models of Toeplitz Operators

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Toeplitz Operators and Spectral Function Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 42))

Abstract

We consider a Toeplitz operator TF whose symbol F is continuous on the unit circle, analytic in the unit disc except for a finite number of poles and has non-negative winding number with respect to any point in C .It is shown that TF is similar to the multiplication operator by the projection II on a certain function space H 2F (б*) on the so-called ultraspectrum б* of TF which is a Riemann surface projecting into the spectrum б(TF). The ultraspectrum б* and the space H 2F (б*) are evaluated explicitly. As a consequence of this theorem, we describe the commutant, invariant and hyperinvariant subspaces of TF for a wide class of symbols. In particular, suppose that F has a finite number of selfintersections and some additional conditions hold. We show that in this case the commutant of TF is isomorphic to H∞ (бc) for a Riemann surface бc and hence is abelian.

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Authors
  1. D. V. Yakubovich
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N. K. Nikolskii

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© 1989 Springer Basel AG

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Yakubovich, D.V. (1989). Riemam Surface Models of Toeplitz Operators. In: Nikolskii, N.K. (eds) Toeplitz Operators and Spectral Function Theory. Operator Theory: Advances and Applications, vol 42. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5587-7_7

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  • DOI: https://doi.org/10.1007/978-3-0348-5587-7_7

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5589-1

  • Online ISBN: 978-3-0348-5587-7

  • eBook Packages: Springer Book Archive

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