Pointwise multipliers between spaces of analytic functions

Abstract

A Banach space X of analytic function in , the unit disc in , is said to be admissible if it contains the polynomials and convergence in X implies uniform convergence in compact subsets of . If X and Y are two admissible Banach spaces of analytic functions in and g is a holomorphic function in , g is said to be a multiplier from X to Y if g ·f ∈ Y for every f ∈ 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 ∈ M (X, Y) then the multiplication operator M g, defined by Mg (f) = g · f, is a bounded operator from X into Y. It is known that M (X) ⊂ H ∞ and that if g ∈ M (X), then ∥g∥H ∞ ≤ ||Mg||. Clearly, this implies that M (X, Y) ⊂ H ∞ if Y ⊂ X. If Y ⊄ X, the inclusion M (X, Y) ⊂ H ∞ 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) ⊂ H ∞ holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Qs-spaces (0<s<∞).