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.


47B33 Linear composition operators


Zbl 1112.47019
Full Text: DOI EuDML


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