Semigroups of holomorphic self-maps of domains and one-parameter semigroups of isometries of Bergman spaces
A holomorphic self-map \(\Phi\) of a domain \(\Omega\subseteq \mathbb{C}^N\) gives rise to a linear operator \(C_\Phi:H(\Omega,\mathbb{C})\to H(\Omega,\mathbb{C})\), \(C_\Phi f=f\circ \Phi\). Such composition operators were already extensively studied and are also investigated in this paper. First, the relationship between the equicontinuity and precompactness of families \(\mathcal{F}\subseteq H(\Omega,\Omega)\) is explored. It turns out that \(\Omega\) is a domain of holomorphy if and only if equicontinuity and precompactness are equivalent in \(H(\Omega,\Omega)\). Given a (continuous) semigroup \(\Phi:[0,+\infty)\to H(\Omega,\Omega)\), the associated one-parameter semigroup \(\{C_{\Phi_t}\}_{t\geq 0}\) of composition operators on the space \(H(\Omega,\mathbb{C})\) is investigated. The infinitesimal generator is identified and the unique solvability of the associated initial value problem is established. By means of these results on the linear semigroup \(\{C_{\Phi_t}\}_{t\geq 0}\), the following previously known facts concerning \(\Phi_t\) are easily reproduced: \(\Phi_t\) is always univalent [see \textit{M.~Abate}, Ann.~Mat.~Pura Appl., IV.~Ser.~161, 167--180 (1992; Zbl 0758.32013)] and the mapping \(\Phi(t,z)=\Phi_t(z)\) is jointly \(C^\infty\), real analytic in \(t\) and holomorphic in \(z\). For a semigroup \(\{T_t\}_{t\geq0}\) of isometries on the Bergman space \(L^p_a(\Omega)\) (\(1\leq p<+\infty\), \(p\neq 2\)), where \(\Omega\) is a simply connected, (bounded) Runge domain, there always exists a semigroup \(\Phi\) of full holomorphic self-maps of \(\Omega\) such that \(T_t\) is a weighted composition operator associated with \(\Phi_t\) with an appropriate, continuous weight. The core of the proof is the characterisation of isometries on Bergman spaces by \textit{C.~Kolaski} [Can.~J.~Math.~33, 1157--1164 (1981; Zbl 0487.32002)]. On the other hand, each such semigroup of weighted composition operators is a strongly continuous semigroup on \(L^p_a(\Omega)\) also for \(p=2\). If \(p\neq 2\), on explicit formula is given for the infinitesimal generator of \(\{T_t\}_{t\geq0}\), and by using general one-parameter semigroup theory, spectral properties of the semigroup \(\{T_t\}_{t\geq0}\) and its infinitesimal generator are studied. Furthermore, some investigations are carried out in the \(p=2\) case for \(L_a^p(\Omega)\) and the semigroup \(\{T_t\}_{t\geq0}\) is also considered on \(L^2(\Omega)\).
scientific article; zbMATH DE number 1997314
scientific article