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Proceedings of the American Mathematical Society

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

 

 

Analytic Toeplitz operators with automorphic symbol


Author: M. B. Abrahamse
Journal: Proc. Amer. Math. Soc. 52 (1975), 297-302
MSC: Primary 47B35
MathSciNet review: 0405156
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Abstract: Let $ R$ denote the annulus $ \{ z:1/2 < \vert z\vert < 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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DOI: https://doi.org/10.1090/S0002-9939-1975-0405156-8
Keywords: Toeplitz operator, automorphic function, universal covering map
Article copyright: © Copyright 1975 American Mathematical Society