Multipliers and Hadamard products in the vector-valued setting

Abstract

Let Ei be Banach spaces, and let XEi be Banach spaces continuously contained in the spaces of Ei-valued sequences (xˆ(j))j∈EiN, for i=1,2,3. Given a bounded bilinear map B:E1×E2→E3, we define (XE2,XE3)B, the space of B-multipliers between XE2 and XE3, to be the set of sequences (λj)j∈E1N such that (B(λj,xˆ(j)))j∈XE3 for all (xˆ(j))j∈XE2, and we define the Hadamard projective tensor product XE1⊛BXE2 as consisting of those elements in E3N that can be represented as ∑n∑jB(xˆn(j),yˆn(j)), where (xn)n∈XE1, (yn)n∈XE2, and ∑n‖xn‖XE1‖yn‖XE2<∞. We will analyze some properties of these two spaces, relate them, and compute the Hadamard tensor products and the spaces of vector-valued multipliers in several cases, getting applications in the particular case where E=L(E1,E2) and B(T,x)=T(x).