×

Spectrum of compact weighted composition operators on the weighted Hardy space in the unit ball. (English) Zbl 1138.47021

The paper extends to the unit ball \(B_n\) of \({\mathbb C}^n\) the results by G. Gunatillake [Proc. Am. Math. Soc. 135, No. 2, 461–467 (2007; Zbl 1112.47019)] in the unit disc concerning the spectrum of weighted composition operators \(C_{\psi,\varphi}(f)(z)=\psi(z)f(\varphi(z))\) on the weighted Hardy space. The main theorem establishes that if \(C_{\psi,\varphi}\) is a compact operator on \(H^2(\beta,B_n)\) and \(\varphi\) has only one fixed point \(a\) in the unit ball, then the spectrum of \(C_{\psi,\varphi}\) is the set \(\{0,\psi(a),\psi(a)\mu\}\), where \(\mu\) denotes all possible products of the eigenvalues of \(\varphi'(a)\). Then the authors find conditions on \(\psi\) and \(\varphi\) to be able to apply their result.

MSC:

47B33 Linear composition operators

Citations:

Zbl 1112.47019
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, Fla, USA; 1995:xii+388.
[2] Gunatillake, G, Spectrum of a compact weighted composition operator, Proceedings of the American Mathematical Society, 135, 461-467, (2007) · Zbl 1112.47019
[3] Aron, R; Lindström, M, Spectra of weighted composition operators on weighted Banach spaces of analytic functions, Israel Journal of Mathematics, 141, 263-276, (2004) · Zbl 1074.47010
[4] Clahane, DD, Spectra of compact composition operators over bounded symmetric domains, Integral Equations and Operator Theory, 51, 41-56, (2005) · Zbl 1081.47028
[5] MacCluer, BD; Weir, RJ, Linear-fractional composition operators in several variables, Integral Equations and Operator Theory, 53, 373-402, (2005) · Zbl 1121.47017
[6] Cowen, CC; MacCluer, BD, Linear fractional maps of the ball and their composition operators, Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum, 66, 351-376, (2000) · Zbl 0970.47011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.