Convolution of three functions by means of bilinear maps and applications

Abstract

When dealing with spaces of vector-valued analytic functions there is a natural way to understand multipliers between them. If X and Y are Banach spaces and L(X,Y ) stands for the space of linear and continuous operators we may consider the convolution of L(X,Y )-valued analytic functions, say F (z) = ∑ n=0∞ Tnz , and X-valued polynomials, say f(z) = ∑m n=0 xnz , to get the Y -valued function F ∗ f(z) = ∑ Tn(xn)z. The second author considered such a definition and studied multipliers between H(X) and BMOA(Y ) in [5]. When the functions take values in a Banach algebra A then the natural extension of multiplier is simply that if f(z) = ∑ anz n and g(z) = ∑ bnz , then f ∗ g(z) = ∑ an.bnz n where a.b stands for the product in the algebra A. Of course, similarly one can consider an ∈ L(R), bn ∈ L(R) and the convolution an ∗ bn ∈ L(R) (where p, q, r verifies the condition on Young’s theorem). The reader is referred to [3] for results along these lines. In this paper we shall consider a much more general notion of convolutions coming from general bilinear maps and that will extend the previous examples. Assume X,Y, Z are Banach spaces and let u : X × Y → Z be a bounded bilinear map. Given a X-valued polynomial f(z) = ∑m n=0 xnz n and given a Y -valued polynomial g(z) = ∑k n=0 ynz n we define the u-convolution of f an g as the polynomial given by