The taylor coefficient problem for banach-and hilbert-space-valued bounded analytic functions

Abstract

A classical I. Schur algorithm describing the Taylor coefficient sequences for bounded in the unit disk analytic functions is generalized to the case when functions take on values in a complex Banach space E. Let H ∞(E) be the space of E-valued bounded analytic in the unit disk functions; ρ H ∞(E)→E n be the mapping that maps each f ∊ H ∞(E) to its first n taylor coefficients; B (E n ) be the p-image in E of the closed unit bail in H ∞(E). The description of the boundary and interior point of the set B(E n ) in a term of a recursion process (a generalized Schur algorithm) is given for the case when the group of diffeomorphisms acts transitively inside the unit ball of Banach space E. A case of an arbitrary infinite-dimensional complex Hilbert space of function values also has been considered. The theorem and its corollary are presented which make it possible to reduce the problem lo the corresponding one for bounded analytic functions taking on values in a finite-dimensional Euclidean space, the latter having been completely solved in [2, 3, 7].