The strict topology on spaces of bounded holomorphic functions.

Abstract

The authors consider the space of bounded holomorphic functions on the open unit ball of a Banach space with the natural analogue of the strict topology β for the one- (or finite-)dimensional case. This can be defined by means of weighted seminorms or as the mixed topology (in the sense of Wiweger) of the supremum norm and the topology of uniform convergence on subsets of the unit ball which are bounded away from its complement. The natural analogues of some elementary properties of the one-dimensional case are obtained. In a second section, some properties of H∞ as a topological algebra are discussed. In particular, the spectrum is identified (under some rather restrictive conditions on the Banach space) and this is used to obtain a representation for strictly continuous homomorphisms between such H∞-algebras