POINTWISE MULTIPLIERS BETWEEN SPACES OF ANALYTIC FUNCTIONS

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 center dot f is an element of Y for every f is an element of 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 an element of M (X; Y) then the multiplication operator M-g, defined by M-g (f) = g center dot f, is a bounded operator from X into Y. It is known that M (X) subset of H-infinity and that if g is an element of M(X), then parallel to g parallel to H-infinity <= parallel to M-g parallel to Clearly, this implies that M (X; Y) subset of H-infinity if Y subset of X. If Y not subset of X, the inclusion M (X; Y) subset of H-infinity 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) subset of H-infinity holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Q(s)-spaces (0 < s < infinity).