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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Analytic Toeplitz operators with automorphic symbol
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by M. B. Abrahamse PDF
Proc. Amer. Math. Soc. 52 (1975), 297-302 Request permission

Abstract:

Let $R$ denote the annulus $\{ z:1/2 < |z| < 1\}$ and let $\pi$ be a holomorphic universal covering map from the unit disk onto $R$. It is shown that if $\pi$ is a function of an inner function $\omega$, that is, if $\pi (z) = \pi (\omega (z))$, then $\omega$ is a linear fractional transformation. However, the analytic Toeplitz operator ${T_\pi }$ has nontrivial reducing subspaces. These facts answer in the negative a question raised by Nordgren [10]. Let $\phi$ be the function $\phi (z) = \pi (z) - 3/4$ and let $\phi = \chi F$ be the inner-outer factorization of $\phi$. An operator $C$ is produced which commutes with ${T_\phi }$ but does not commute with ${T_\chi }$ nor with ${T_F}$. This answers in the negative a question raised by Deddens and Wong [7]. The functions $\pi$ and $\phi$ are both automorphic under the group of covering transformations for $\pi$ and hence may be viewed as functions on the annulus $R$. This point of view is critical in these examples.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 52 (1975), 297-302
  • MSC: Primary 47B35
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0405156-8
  • MathSciNet review: 0405156