Spectrum of a compact weighted composition operator
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- by Gajath Gunatillake PDF
- Proc. Amer. Math. Soc. 135 (2007), 461-467 Request permission
Abstract:
For $\psi$ analytic in the open unit disk and $\varphi$ an analytic map from the unit disk into itself, the weighted composition operator $C_{\psi ,\varphi }$ is the operator on the weighted Hardy space $H^{2}(\beta )$ given by $(C_{\psi ,\varphi }f)(z)=\psi (z)f(\varphi (z)).$ This paper discusses the spectrum of $C_{\psi ,\varphi }$ when it is compact on a certain class of weighted Hardy spaces and when the composition map $\varphi$ has a fixed point inside the open unit disk.References
- Paul S. Bourdon, David Levi, Sivaram K. Narayan, and Joel H. Shapiro, Which linear-fractional composition operators are essentially normal?, J. Math. Anal. Appl. 280 (2003), no. 1, 30–53. MR 1972190, DOI 10.1016/S0022-247X(03)00005-2
- John H. Clifford and Michael G. Dabkowski, Singular values and Schmidt pairs of composition operators on the Hardy space, J. Math. Anal. Appl. 305 (2005), no. 1, 183–196. MR 2128121, DOI 10.1016/j.jmaa.2004.11.014
- 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
- Frank Forelli, The isometries of $H^{p}$, Canadian J. Math. 16 (1964), 721–728. MR 169081, DOI 10.4153/CJM-1964-068-3
- G. Gunatillake, Weighted Composition Operators, Doctoral Dissertation, Purdue University, 2005.
- C. Hammond, On the Norm of a Composition Operator, Doctoral Dissertation, University of Virginia, 2003.
- Joel H. Shapiro and Wayne Smith, Hardy spaces that support no compact composition operators, J. Funct. Anal. 205 (2003), no. 1, 62–89. MR 2020208, DOI 10.1016/S0022-1236(03)00215-5
Additional Information
- Gajath Gunatillake
- Affiliation: Department of Mathematics and Statistics, American University of Sharjah, P. O. Box 26666, Sharjah, United Arab Emirates
- Received by editor(s): February 3, 2005
- Received by editor(s) in revised form: September 19, 2005
- Published electronically: September 11, 2006
- Communicated by: Joseph A. Ball
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 461-467
- MSC (2000): Primary 47B32
- DOI: https://doi.org/10.1090/S0002-9939-06-08497-8
- MathSciNet review: 2255292