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 ∈  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 M_g(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^∞ ≤ ∥M_g∥. 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 Q_s-spaces (0 < s < ∞ ).