The commutant of an analytic Toeplitz operator
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- by Carl C. Cowen PDF
- Trans. Amer. Math. Soc. 239 (1978), 1-31 Request permission
Abstract:
For a function f in ${H^\infty }$ of the unit disk, the operator on ${H^2}$ of multiplication by f will be denoted by ${T_f}$ and its commutant by $\{ {T_f}\} ’$. For a finite Blaschke product B, a representation of an operator in ${\{ {T_B}\}’}$ as a function on the Riemann surface of ${B^{ - 1}} \circ B$ motivates work on more general functions. A theorem is proved which gives conditions on a family $\mathcal {F}$ of ${H^\infty }$ functions which imply that there is a function h such that $\{ {T_h}\} ’ = { \cap _{f \in \mathcal {F}}}\{ {T_f}\} ’$. As a special case of this theorem, we find that if the inner factor of $f - f(c)$ is a finite Blaschke product for some c in the disk, then there is a finite Blaschke product B with $\{ {T_f}\} ’ = \{ {T_B}\} ’$. Necessary and sufficient conditions are given for an operator to commute with ${T_f}$ when f is a covering map (in the sense of Riemann surfaces). If f and g are in ${H^\infty }$ and $f = h \circ g$, then $\{ {T_f}\} ’ \supset \{ {T_g}\} ’$. This paper introduces a class of functions, the ${H^2}$-ancestral functions, for which the converse is true. If f and g are ${H^2}$-ancestral functions, then $\{ {T_f}\} ’ \ne \{ {T_g}\} ’$ unless $f = h \circ g$ where h is univalent. It is shown that inner functions and covering maps are ${H^2}$-ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function f which imply that ${T_f}$ does not commute with nonzero compact operators. It follows from one of these results that if f is an ${H^2}$-ancestral function, then ${T_f}$ does not commute with any nonzero compact operators.References
- M. B. Abrahamse, Analytic Toeplitz operators with automorphic symbol, Proc. Amer. Math. Soc. 52 (1975), 297–302. MR 405156, DOI 10.1090/S0002-9939-1975-0405156-8
- 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 454714, DOI 10.1090/S0002-9939-1976-0454714-4
- 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 355046, DOI 10.1215/S0012-7094-74-04177-5
- James A. Deddens and Tin Kin Wong, The commutant of analytic Toeplitz operators, Trans. Amer. Math. Soc. 184 (1973), 261–273. MR 324467, DOI 10.1090/S0002-9947-1973-0324467-0
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Eric A. Nordgren, Composition operators, Canadian J. Math. 20 (1968), 442–449. MR 223914, DOI 10.4153/CJM-1968-040-4
- Carl Pearcy and Allen L. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 219–229. MR 0355639
- Walter Rudin, A generalization of a theorem of Frostman, Math. Scand. 21 (1967), 136–143 (1968). MR 235151, DOI 10.7146/math.scand.a-10853
- John V. Ryff, Subordinate $H^{p}$ functions, Duke Math. J. 33 (1966), 347–354. MR 192062
- H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532. MR 133446, DOI 10.2307/2372892
- A. L. Shields and L. J. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J. 20 (1970/71), 777–788. MR 287352, DOI 10.1512/iumj.1971.20.20062
- James E. Thomson, Intersections of commutants of analytic Toeplitz operators, Proc. Amer. Math. Soc. 52 (1975), 305–310. MR 399927, DOI 10.1090/S0002-9939-1975-0399927-4
- James E. Thomson, The commutants of certain analytic Toeplitz operators, Proc. Amer. Math. Soc. 54 (1976), 165–169. MR 388156, DOI 10.1090/S0002-9939-1976-0388156-7
- James E. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math. 99 (1977), no. 3, 522–529. MR 461196, DOI 10.2307/2373929
- James Thomson, The commutant of a class of analytic Toeplitz operators. II, Indiana Univ. Math. J. 25 (1976), no. 8, 793–800. MR 417843, DOI 10.1512/iumj.1976.25.25063
- William A. Veech, A second course in complex analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0220903
- E. L. Stout, Bounded holomorphic functions on finite Reimann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255–285. MR 183882, DOI 10.1090/S0002-9947-1965-0183882-4
Additional Information
- © Copyright 1978 American Mathematical Society
- 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