Composition operators on weighted Hardy spaces (Q470389)

The author considers the weighted Hardy space \({\mathcal H}^{p} (\beta)\), \(1 < p < \infty\), consisting of functions analytic on the open unit disc \(\mathbb{D}\), where the weight sequence \(\beta := \{\beta(n) \}_n\) of positive numbers satisfies \(\beta(0) = 1\) and \(\liminf_{n \rightarrow \infty} \beta(n)^{1/n} = 1\). The space \({\mathcal H}^{p} (\beta)\) is called small if \(\sum_{n=0}^{\infty} \beta ^{-r}(n) < \infty\), in which \(r\) is the conjugate exponent of \(p\).NEWLINENEWLINEFor an analytic map \(\psi\) on \(\mathbb{D}\) and an analytic self-map \(\varphi\) of \(\mathbb{D}\), the weighted composition operator \(W_{\psi , \varphi}\) is defined on \({\mathcal H}^{p} (\beta)\) by \(W_{\psi , \varphi} f = \psi . (f \circ \varphi )\). The usual composition operator \(C_{\varphi}\) is \(W_{1, \varphi}\).NEWLINENEWLINEIn the paper under review, boundedness and compactness of \(W_{\psi , \varphi}\) are discussed. Furthermore, some conditions are presented under which a composition operator is in the Schatten \(p\)-class, \(p > 0\). To be more precise, Section~1 contains some definitions and preliminary results. In Section~2, some sufficient conditions for the boundedness of \(W_{\psi , \varphi}\) from a small \({\mathcal H}^{p} (\beta_1)\) into \({\mathcal H}^{q} (\beta_2)\) are presented. Then necessary conditions are given for the boundedness of \(W_{\psi , \varphi}\) on a small \({\mathcal H}^{p} (\beta)\). In Section~3, sufficient conditions for the compactness of \(W_{\psi , \varphi}\) are presented. Section~4 is devoted to investigate necessary conditions for the compactness of \(C_{\varphi}\) and more generally, \(W_{\psi , \varphi}\) in terms of the fixed points of \(\varphi\) in \(\mathbb{D}\). Section~5 is concerned with compact composition operators and the behavior of the angular derivative of their inducing maps. The paper concludes with generalizations of the results which were obtained in [\textit{N. Zorboska}, Proc. Am. Math. Soc. 106, No. 3, 679--684 (1989; Zbl 0686.47027)]. Some conditions are established under which an analytic self-map of \(\mathbb{D}\) with the supremum norm strictly less than 1 induces a bounded composition operator in every Schatten \(p\)-class \(S_{p}(H^s(\beta_1),H^q(\beta_2))\), \(p>0\).