Coefficient multipliers on Banach spaces of analytic functions

Motivated by an old paper of Wells [34] we define the space X circle times Y, where X and Y are "homogeneous" Banach spaces of analytic functions on the unit disk D, by the requirement that f can be represented as f = Sigma(infinity)(j=0) g(n)*h(n), with g(n) is an element of X, h(n) is an element of Y and Sigma(infinity)(n=1) parallel to gn parallel to(X)parallel to hn parallel to(Y) < infinity. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula ((X circle times Y), Z) = (X, (Y, Z)), where (U, V) denotes the space of multipliers from U to V, and as a special case (X circle times Y)* = (X, Y*), where U* = (U, H-infinity). We determine H-1 circle times X for a class of spaces that contains H-p and l(p) (1 <= p <= 2), and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.