An operator representation for weighted spaces of vector valued homomorphic functions
Let \(G\) be an open set in \(\mathbb{C}^N\) and \(V\) a Nachbin family of weights on \(G\) such that for each compact set \(K\) in \(G\) there exists \(v\in V\) with \(\inf_{z\in K} v(z)>0\). For a locally convex (l.c.) space \(E\) define the associated weighted space of holomorphic functions as \[ HV(G,E):= \{f: G\to E:f\text{ is holomorphic and \(vf\) is bounded in \(E\;\forall v\in V\)}\}, \] and endow it with the l.c. topology induced by the semi-norms \[ q_{v,p}: f\mapsto \sup\{v(z)p(f(z)): z\in G\},\;p\text{ a continuous seminorm on }E. \] Then the main results of the paper are: For each quasibarrelled l.c. space \(E\) the map \[ \Phi: HV(G, E_b')\to{\mathcal L}_b(E, HV(G,\mathbb{C})),\;\Phi(f): e\mapsto f(\cdot)[e] \] is a linear topological isomorphism. Furthermore, if \(G_j\subset \mathbb{C}^{N_j}\) and \(V_j\) are as above for \(j= 1,2\), then \[ HV_1(G_1, HV_2(G_2))= H(V_1\otimes V_2)(G_1\times G_2). \] Besides this, related results for continuous functions, applications and connections to work of other authors are discussed.
scientific article; zbMATH DE number 1346909
scientific article