Shift-invariant subspaces invariant for composition operators on the Hardy-Hilbert space
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- by Carl C. Cowen and Rebecca G. Wahl PDF
- Proc. Amer. Math. Soc. 142 (2014), 4143-4154 Request permission
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
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Additional Information
- Carl C. Cowen
- Affiliation: Department of Mathematical Sciences, IUPUI (Indiana University–Purdue University, Indianapolis), Indianapolis, Indiana 46202-3216
- MR Author ID: 52315
- Email: ccowen@math.iupui.edu
- Rebecca G. Wahl
- Affiliation: Department of Mathematics, Butler University, Indianapolis, Indiana 46208-3485
- Email: rwahl@butler.edu
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3266985