Multipliers and Hadamard products in the vector-valued setting

Abstract

Let $E_i$ be Banach spaces, and let $X_{E_i}$ be Banach spaces continuously contained in the spaces of $E_i$-valued sequences $\left(\hat{x}\right(j){)}_j\in E_i^{\mathbb{N}}$, for $i=1,2,3$. Given a bounded bilinear map $B:E_1\times E_2\to E_3$, we define $(X_{E_2},X_{E_3}{)}_B$, the space of $B$-multipliers between $X_{E_2}$ and $X_{E_3}$, to be the set of sequences $({\lambda }_j{)}_j\in E_1^{\mathbb{N}}$ such that $\left(B\right({\lambda }_j,\hat{x}\left(j\right)){)}_j\in X_{E_3}$ for all $\left(\hat{x}\right(j){)}_j\in X_{E_2}$, and we define the Hadamard projective tensor product $X_{E_1}{⊛}_B{X}_{E_2}$ as consisting of those elements in $E_3^{\mathbb{N}}$ that can be represented as ${\sum }_n{\sum }_{j}B\left({\hat{x}}_n\right(j),{\hat{y}}_n(j\left)\right)$, where $(x_n{)}_n\in X_{E_1}$, $(y_n{)}_n\in X_{E_2}$, and ${\sum }_n\Vert x_n{\Vert }_{X_{E_1}}\Vert y_n{\Vert }_{X_{E_2}}<\infty$.

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 $E=\mathcal{L}(E_1,E_2)$ and $B(T,x)=T\left(x\right)$.