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Shift-invariant subspaces invariant for composition operators on the Hardy-Hilbert space


Authors: Carl C. Cowen and Rebecca G. Wahl
Journal: Proc. Amer. Math. Soc. 142 (2014), 4143-4154
MSC (2010): Primary 47B33; Secondary 47B38, 47A15
DOI: https://doi.org/10.1090/S0002-9939-2014-12132-0
Published electronically: July 29, 2014
MathSciNet review: 3266985
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Abstract: If $ \varphi $ is an analytic map of the unit disk $ \mathbb{D}$ into itself, the composition operator $ C_{\varphi }$ on a Hardy space $ H^{2}$ is defined by $ C_{\varphi }(f)=f\circ \varphi $. The unilateral shift on $ H^{2}$ is the operator of multiplication by $ z$. Beurling (1949) characterized the invariant subspaces for the shift. In this paper, we consider the shift-invariant subspaces that are invariant for composition operators. More specifically, necessary and sufficient conditions are provided for an atomic inner function with a single atom to be invariant for a composition operator, and the Blaschke product invariant subspaces for a composition operator are described. We show that if $ \varphi $ has Denjoy-Wolff point $ a$ on the unit circle, the atomic inner function subspaces with a single atom at $ a$ are invariant subspaces for the composition operator $ C_{\varphi }$.


References [Enhancements On Off] (What's this?)

  • [1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), no. 1, 239-255, DOI 10.1007/BF02395019. MR 0027954 (10,381e)
  • [2] Carl C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), no. 1, 69-95. MR 607108 (82i:30036), https://doi.org/10.2307/1998482
  • [3] Carl C. Cowen, Composition operators on $ H^{2}$, J. Operator Theory 9 (1983), no. 1, 77-106. MR 695941 (84d:47038)
  • [4] Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026 (97i:47056)
  • [5] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981. MR 628971 (83g:30037)
  • [6] M. M. Jones, Shift invariant subspaces of composition operators on $ H^p$, Arch. Math. (Basel) 84 (2005), no. 3, 258-267. MR 2134140 (2005m:47046), https://doi.org/10.1007/s00013-004-1017-z
  • [7] Ali Mahvidi, Invariant subspaces of composition operators, J. Operator Theory 46 (2001), no. 3, suppl., 453-476. MR 1897149 (2003c:47044)
  • [8] D. J. Newman, Interpolation in $ H^{\infty }$, Trans. Amer. Math. Soc. 92 (1959), 501-507. MR 0117350 (22 #8130)
  • [9] H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513-532. MR 0133446 (24 #A3280)

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Additional Information

Carl C. Cowen
Affiliation: Department of Mathematical Sciences, IUPUI (Indiana University–Purdue University, Indianapolis), Indianapolis, Indiana 46202-3216
Email: ccowen@math.iupui.edu

Rebecca G. Wahl
Affiliation: Department of Mathematics, Butler University, Indianapolis, Indiana 46208-3485
Email: rwahl@butler.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12132-0
Keywords: Composition operator, shift-invariant subspace
Received by editor(s): March 26, 2012
Received by editor(s) in revised form: January 3, 2013
Published electronically: July 29, 2014
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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