The spectra of composition operators from linear fractional maps acting upon the Dirichlet space
Let \(H(\mathbb{D})\) denote the space of all analytic functions in the open unit disk equipped with the topology of ``uniform converge on compact subsets''. The Dirichlet space, denoted by \(\mathcal{D}\), consists of all \(f\in H(\mathbb{D})\) for which \(\int_{\mathbb{D}} | f'| ^2 dA < \infty\), where \(dA\) is the Lebesgue area measure. The norm for \(\mathcal{D}\) is defined by \[ \| f\| ^2_{\mathcal{D}}=| f(0)| ^2+\frac{1}{\pi} \int_{\mathbb{D}} | f'| ^2 dA ,\quad f \in {\mathcal{D}}. \] \textit{E. A. Nordgren} determined the spectrum of the composition operator \(C_{\varphi}\) on the Hardy space \(H^2\) induced from an automorphism \(\varphi\) of \(\mathbb{D}\) [Can. J. Math. 20, 442--449 (1968; Zbl 0161.34703)]. The present author determines the spectrum of \(C_{\varphi}\) as an operator on the Dirichlet space when \(\varphi\) is an automorphism, and, more generally, a linear fractional map on \(\mathbb{D}\). Although eigenfunctions for the operator \(C_{\varphi}\) on \(H^2\) frequently do not belong to the space \(\mathcal{D}\), spectral results hold for the operator \(C_{\varphi}\) on \(\mathcal{D}\) much like those already known for the operator \(C_{\varphi}\) on \(H^2\).
scientific article; zbMATH DE number 2141636