Invertible weighted composition operators
HTML articles powered by AMS MathViewer
- by Paul S. Bourdon PDF
- Proc. Amer. Math. Soc. 142 (2014), 289-299 Request permission
Abstract:
Let $X$ be a set of analytic functions on the open unit disk $\mathbb {D}$, and let $\varphi$ be an analytic function on $\mathbb {D}$ such that $\varphi (\mathbb {D})\subseteq \mathbb {D}$ and $f\mapsto f\circ \varphi$ takes $X$ into itself. We present conditions on $X$ ensuring that if $f\mapsto f\circ \varphi$ is invertible on $X$, then $\varphi$ is an automorphism of $\mathbb {D}$, and we derive a similar result for mappings of the form $f\mapsto \psi \cdot (f\circ \varphi )$, where $\psi$ is some analytic function on $\mathbb {D}$. We obtain as corollaries of this purely function-theoretic work new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as $S^p$ and the weighted Hardy spaces $H^2(\beta )$.References
- Robert F. Allen and Flavia Colonna, Weighted composition operators from $H^\infty$ to the Bloch space of a bounded homogeneous domain, Integral Equations Operator Theory 66 (2010), no. 1, 21–40. MR 2591634, DOI 10.1007/s00020-009-1736-4
- Paul S. Bourdon, Spectra of some composition operators and associated weighted composition operators, J. Operator Theory 67 (2012), no. 2, 537–560. MR 2928328
- 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
- Eva A. Gallardo-Gutiérrez and Jonathan R. Partington, The role of BMOA in the boundedness of weighted composition operators, J. Funct. Anal. 258 (2010), no. 11, 3593–3603. MR 2606866, DOI 10.1016/j.jfa.2010.02.019
- Gajath Gunatillake, Compact weighted composition operators on the Hardy space, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2895–2899. MR 2399056, DOI 10.1090/S0002-9939-08-09247-2
- Gajath Gunatillake, Spectrum of a compact weighted composition operator, Proc. Amer. Math. Soc. 135 (2007), no. 2, 461–467. MR 2255292, DOI 10.1090/S0002-9939-06-08497-8
- Gajath Gunatillake, Invertible weighted composition operators, J. Funct. Anal. 261 (2011), no. 3, 831–860. MR 2799582, DOI 10.1016/j.jfa.2011.04.001
- Barbara D. MacCluer, Composition operators on $S^p$, Houston J. Math. 13 (1987), no. 2, 245–254. MR 904956
- Barbara D. MacCluer, Xiangfei Zeng, and Nina Zorboska, Composition operators on small weighted Hardy spaces, Illinois J. Math. 40 (1996), no. 4, 662–677. MR 1415024
- Barbara D. MacCluer, Fredholm composition operators, Proc. Amer. Math. Soc. 125 (1997), no. 1, 163–166. MR 1371134, DOI 10.1090/S0002-9939-97-03743-X
- Valentin Matache, Weighted composition operators on $H^2$ and applications, Complex Anal. Oper. Theory 2 (2008), no. 1, 169–197. MR 2390678, DOI 10.1007/s11785-007-0025-y
- Raymond C. Roan, Composition operators on the space of functions with $H^{p}$-derivative, Houston J. Math. 4 (1978), no. 3, 423–438. MR 512878
- Joel H. Shapiro, Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 1, 49–57. MR 883400, DOI 10.1090/S0002-9939-1987-0883400-9
- Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
- Gerald D. Taylor, Multipliers on $D_{\alpha }$, Trans. Amer. Math. Soc. 123 (1966), 229–240. MR 206696, DOI 10.1090/S0002-9947-1966-0206696-6
Additional Information
- Paul S. Bourdon
- Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
- Email: psbourdon@gmail.com
- Received by editor(s): September 12, 2011
- Received by editor(s) in revised form: March 11, 2012
- Published electronically: October 3, 2013
- Communicated by: Richard Rochberg
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 289-299
- MSC (2010): Primary 47B33, 30J99
- DOI: https://doi.org/10.1090/S0002-9939-2013-11804-6
- MathSciNet review: 3119203