Algebras of holomorphic functions
Let \(H(U)\) denote the algebra of holomorphic functions on an open subset \(U\) of a complex locally convex Hausdorff space \(E\) and let \(H_ c(U)\) and \(H_ e(U)\) denote this algebra when supplied with the continuous and the associated equable convergence structures, respectively. One objective of this note is to show that the convergence algebras \(H_ c(U)\) and \(H_ e(U)\) in some (quite general) cases contain total information concerning the space \(U\) in the sense that the spectrum of \(H_ c(U)\) or \(H_ e(U)\) is homeomorphic to \(U\), when it is given the continuous convergence structure. The finest locally convex topologies on \(H(U)\) coarser than the convergence structures of \(H_ c(U)\) and \(H_ e(U)\) will be denoted by \(\kappa\) and \(\lambda\). In [Monatsh. Math. 101, 11-26 (1986; Zbl 0579.46031)] we proved that \(H_{\lambda}(U)\) coincides with \(H_{\tau_{\delta}}(U)\) if \(U\) is a Lindelöf space and that \(\lambda\) is always finer than the ported topology \(\tau_{\omega}\). Isidoro and Mujica have characterized the spectrum (as a set) of \(H_{co}(U)\), \(H_{\tau_{\omega}}(U)\), and \(H_{\tau_{\delta}}(U)\) and it has been used, e.g., for constructing the envelope of holomorphy of \(U\). Using the result about the spectrum of \(H_ c(U)\) and \(H_ e(U)\), mentioned above, we obtain information on the spectrum of \(H_{\kappa}(U)\) and \(H_{\lambda}(U)\). Finally, we prove that the bornological topology associated with the continuous convergence structure on the vector space \(H(U,F)\), of holomorphic functions \(U\to F\), coincides with the compact- open topology, when \(U\) is an open subset of a DFM-space \(E\) and \(F\) is a metrizable locally convex space. This result is analogous to a similar result of B. Müller concerning spaces of continuous linear functions.
algebra of holomorphic functions on an open subset of a complex locally convex Hausdorff space
associated equable convergence structures
bornological topology associated with the continuous convergence structure
scientific article; zbMATH DE number 4078028
scientific article