Pointwise multipliers between spaces of analytic functions.

Abstract

A Banach space X of analytic function in D, the unit disc in C, is said to be admissible if it contains the polynomials and convergence in X implies uniform convergence in compact subsets of D. If X and Y are two admissible Banach spaces of analytic functions in D and g is a holomorphic function in D, g is said to be a multiplier from X to Y if g · f is in Y for every f in X. The space of all multipliers from X to Y is denoted M(X; Y ), and M(X) will stand for M(X;X). The closed graph theorem shows that if g is in M(X; Y ) then the multiplication operator Mg, defi ned by Mg(f) = g · f, is a bounded operator from X into Y. It is known that M(X) c H^inf and that if g is in M(X), then ∥g∥_H^inf <= ∥Mg∥. Clearly, this implies that M(X; Y ) c H^inf if Y c X. If Y is not contained in X, the inclusion M(X; Y ) c H^inf may not be true. In this paper we start presenting a number of conditions on the spaces X and Y which imply that the inclusion M(X; Y ) c H^inf holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Qs-spaces (0 < s < 1).