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 | References | Similar Articles | Additional Information
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.
- [1] M. B. Abrahamse, Toeplitz operators in multiply connected regions, Bull. Amer. Math. Soc. 77 (1971), 449–454. MR 273435, https://doi.org/10.1090/S0002-9904-1971-12734-9
- [2] M. B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains, Advances in Math. 19 (1976), no. 1, 106–148. MR 397468, https://doi.org/10.1016/0001-8708(76)90023-2
- [3] M. B. Abrahamse and Thomas L. Kriete, The spectral multiplicity of a multiplication operator, Indiana Univ. Math. J. 22 (1972/73), 845–857. MR 320797, https://doi.org/10.1512/iumj.1973.22.22072
- [4] Lars V. Ahlfors, Bounded analytic functions, Duke Math. J. 14 (1947), 1–11. MR 21108
- [5] I. N. Baker, J. A. Deddens and J. L. Ullman, Entire Toeplitz operators (to appear).
- [6] Joseph A. Ball, Hardy space expectation operators and reducing subspaces, Proc. Amer. Math. Soc. 47 (1975), 351–357. MR 358421, https://doi.org/10.1090/S0002-9939-1975-0358421-7
- [7] James A. Deddens and Tin Kin Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973), 261–273. MR 324467, https://doi.org/10.1090/S0002-9947-1973-0324467-0
- [8] R. G. Douglas and Carl Pearcy, Spectral theory of generalized Toeplitz operators, Trans. Amer. Math. Soc. 115 (1965), 433–444. MR 199706, https://doi.org/10.1090/S0002-9947-1965-0199706-5
- [9] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
- [10] Eric A. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175–181. MR 216321
- [11] Donald Sarason, The 𝐻^{𝑝} spaces of an annulus, Mem. Amer. Math. Soc. No. 56 (1965), 78. MR 0188824
- [12] Michael Voichick, Ideals and invariant subspaces of analytic functions, Trans. Amer. Math. Soc. 111 (1964), 493–512. MR 160920, https://doi.org/10.1090/S0002-9947-1964-0160920-5
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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