We show that a Riemann domain Ω over a symmetrically regular Banach space E admits holomorphic extension to pseudo-convex domains over E with respect to two natural spaces of holomorphic functions of bounded type on Ω. 2004 Elsevier Inc. All rights reserved.There are many classical results in several complex variables theory on the construction of the envelope of holomorphy of a domain Ω ⊂ C n with respect to an subalgebra A ⊂ H (Ω), the set of all holomorphic functions on Ω. Similar results have been obtained in the Banach space setting but mainly for H (Ω). In examining the spectrum of H b (Ω), the space of holomorphic functions of bounded type on the open subset U of the Banach space E, Aron, Galindo, García, and Maestre show that S b (U ), the spectrum of H b (Ω), could be endowed with the structure of a Riemann domain over E when E is symmetrically regular [2]. We continue this line of investigation but found it convenient to begin with a Riemann domain Ω. We show that S b (Ω) is a domain of holomorphy with respect to the space of all holomorphic functions when the underlying space E is symmetrically regular. The question of whether or not it is also a domain of holomorphy with respect to H b (Ω)
We show that the existence of a right inverse at each point for a holomorphic mapping from a pseudoconvex domain in a Banach space with an unconditional basis into a unital Banach algebra implies the existence of a holomorphic right inverse. Variations of this result are given.
We define global Schauder decompositions of locally convex spaces and prove a necessary and sufficient condition for two spaces with global Schauder decompositions to be isomorphic. These results are applied to spaces of entire functions on a locally convex space.Given two spaces, E and F , with Schauder (or even S -absolute) decompositions, the existence of isomorphisms between the spaces forming the decompositions does not imply that E and F are isomorphic. In order to tackle this problem when the underlying decompositions consist of Banach spaces, P. Galindo, M. Maestre and P. Rueda defined in [12] a subclass of S -absolute decompositions of Fréchet spaces: R-Schauder decompositions. To consider the corresponding problem when E and F are locally convex spaces and the underlying decompositions are not necessarily Banach spaces, we were led to define global Schauder decompositions.
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