The commutant of an analytic Toeplitz operator
Author:
Carl C. Cowen
Journal:
Trans. Amer. Math. Soc. 239 (1978), 1-31
MSC:
Primary 47B35; Secondary 30A78
DOI:
https://doi.org/10.1090/S0002-9947-1978-0482347-9
MathSciNet review:
0482347
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For a function f in of the unit disk, the operator on
of multiplication by f will be denoted by
and its commutant by
. For a finite Blaschke product B, a representation of an operator in
as a function on the Riemann surface of
motivates work on more general functions. A theorem is proved which gives conditions on a family
of
functions which imply that there is a function h such that
. As a special case of this theorem, we find that if the inner factor of
is a finite Blaschke product for some c in the disk, then there is a finite Blaschke product B with
. Necessary and sufficient conditions are given for an operator to commute with
when f is a covering map (in the sense of Riemann surfaces). If f and g are in
and
, then
. This paper introduces a class of functions, the
-ancestral functions, for which the converse is true. If f and g are
-ancestral functions, then
unless
where h is univalent. It is shown that inner functions and covering maps are
-ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function f which imply that
does not commute with nonzero compact operators. It follows from one of these results that if f is an
-ancestral function, then
does not commute with any nonzero compact operators.
- [1] M. B. Abrahamse, Analytic Toeplitz operators with automorphic symbol, Proc. Amer. Math. Soc. 52 (1975), 297–302. MR 0405156, https://doi.org/10.1090/S0002-9939-1975-0405156-8
- [2] M. B. Abrahamse and Joseph A. Ball, Analytic Toeplitz operators with automorphic symbol. II, Proc. Amer. Math. Soc. 59 (1976), no. 2, 323–328. MR 0454714, https://doi.org/10.1090/S0002-9939-1976-0454714-4
- [3] I. N. Baker, James A. Deddens, and J. L. Ullman, A theorem on entire functions with applications to Toeplitz operators, Duke Math. J. 41 (1974), 739–745. MR 0355046
- [4] James A. Deddens and Tin Kin Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973), 261–273. MR 0324467, https://doi.org/10.1090/S0002-9947-1973-0324467-0
- [5] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
- [6] Eric A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442–449. MR 0223914, https://doi.org/10.4153/CJM-1968-040-4
- [7] Carl Pearcy and Allen L. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 219–229. Math. Surveys, No. 13. MR 0355639
- [8] Walter Rudin, A generalization of a theorem of Frostman, Math. Scand 21 (1967), 136–143 (1968). MR 0235151, https://doi.org/10.7146/math.scand.a-10853
- [9] John V. Ryff, Subordinate 𝐻^{𝑝} functions, Duke Math. J. 33 (1966), 347–354. MR 0192062
- [10] H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532. MR 0133446, https://doi.org/10.2307/2372892
- [11] A. L. Shields and L. J. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J. 20 (1970/1971), 777–788. MR 0287352, https://doi.org/10.1512/iumj.1971.20.20062
- [12] James E. Thomson, Intersections of commutants of analytic Toeplitz operators, Proc. Amer. Math. Soc. 52 (1975), 305–310. MR 0399927, https://doi.org/10.1090/S0002-9939-1975-0399927-4
- [13] James E. Thomson, The commutants of certain analytic Toeplitz operators, Proc. Amer. Math. Soc. 54 (1976), 165–169. MR 0388156, https://doi.org/10.1090/S0002-9939-1976-0388156-7
- [14] James E. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math. 99 (1977), no. 3, 522–529. MR 0461196, https://doi.org/10.2307/2373929
- [15] James Thomson, The commutant of a class of analytic Toeplitz operators. II, Indiana Univ. Math. J. 25 (1976), no. 8, 793–800. MR 0417843, https://doi.org/10.1512/iumj.1976.25.25063
- [16] William A. Veech, A second course in complex analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0220903
- [17] E. L. Stout, Bounded holomorphic functions on finite Reimann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255–285. MR 0183882, https://doi.org/10.1090/S0002-9947-1965-0183882-4
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1978-0482347-9
Keywords:
Toeplitz operator,
commutant,
,
analytic function,
inner function,
universal covering map
Article copyright:
© Copyright 1978
American Mathematical Society