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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 $ 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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  • [1] M. B. Abrahamse, Toeplitz operators in multiply connected regions, Bull. Amer. Math. Soc. 77 (1971), 449-454. MR 42 #8313. MR 0273435 (42:8313)
  • [2] M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply connected regions, Advances in Math. (to appear). MR 0397468 (53:1327)
  • [3] M. B. Abrahamse and T. Kriete, The spectral multiplicity of a multiplication operator, Indiana J. Math. 22 (1973), 845-857. MR 0320797 (47:9331)
  • [4] L. V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1-11. MR 9, 24. MR 0021108 (9:24a)
  • [5] I. N. Baker, J. A. Deddens and J. L. Ullman, Entire Toeplitz operators (to appear).
  • [6] J. Ball, Hardy space expectation operators and reducing subspaces (preprint). MR 0358421 (50:10887)
  • [7] J. A. Deddens and T. K. Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 186 (1973), 261-273. MR 0324467 (48:2819)
  • [8] R. G. Douglas and C. Pearcy, Spectral theory of generalized Toeplitz operators, Trans. Amer. Math. Soc. 115 (1965), 433-444. MR 33 #7849. MR 0199706 (33:7849)
  • [9] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR 24 #A2844. MR 0133008 (24:A2844)
  • [10] E. A. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175-181. MR 35 #7155. MR 0216321 (35:7155)
  • [11] D. Sarason, The $ {H^p}$ spaces of an annulus, Mem. Amer. Math. Soc. No. 56 (1965). MR 32 #6256. MR 0188824 (32:6256)
  • [12] M. Voichick, Ideals and invariant subspaces of analytic functions, Trans. Amer. Math. Soc. 111 (1964), 493-512. MR 28 #4129. MR 0160920 (28:4129)

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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

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