Weighted composition operators on weighted locally convex spaces of analytic functions (Q1775350)

Let \(V\) be an arbitrary system of weights on an open connected subset \(G\) of \(\mathbb{C}^N\) \((N>1)\). The weighted locally convex spaces of analytic functions, \(HV_b(G)\) and \(HV_0(G)\), with topology generated by seminorms which are weighted analogues of the supremum norm, are defined as \[ HV_b(G)=\{f\in H(G): vf \text{ is bounded on } G \text{ for each }v\in V\} \] and \[ HV_0(G)=\{f\in H(G): vf \text{ vanishes at infinity on }G\text{ for every }v\in V\}. \] The weighted composition operators induced by the analytic mappings \(\phi: G\rightarrow G\) and \(\pi: G\rightarrow \mathbb{C}\) and the invertible weighted composition operators on the spaces \(HV_b(G)\) and \(HV_0(G)\) for different systems of weights \(V\) on \(G\) are characterized in the present paper. Some examples are given to illustrate these theorems.