Analytic Toeplitz operators with automorphic symbol

Author:
M. B. Abrahamse

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

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Abstract: Let denote the annulus and let be a holomorphic universal covering map from the unit disk onto . It is shown that if is a function of an inner function , that is, if , then is a linear fractional transformation. However, the analytic Toeplitz operator has nontrivial reducing subspaces. These facts answer in the negative a question raised by Nordgren [10]. Let be the function and let be the inner-outer factorization of . An operator is produced which commutes with but does not commute with nor with . This answers in the negative a question raised by Deddens and Wong [7]. The functions and are both automorphic under the group of covering transformations for and hence may be viewed as functions on the annulus . This point of view is critical in these examples.

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Additional Information

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