Operator space valued Hankel matrices

Abstract

If $E$ is an operator space, the non-commutative vector valued $L^p$ spaces $S^p[E]$ have been defined by Pisier for any $1 \leq p \leq \infty$. In this paper a necessary and sufficient condition for a Hankel matrix of the form $(a_{i+j})_{0 \le i,j}$ with $a_k \in E$ to be bounded in $S^p[E]$ is established. This extends previous results of Peller where $E=\C$ or $E=S^p$. The main theorem states that if $1 \leq p < \infty$, $(a_{i+j})_{0 \le i,j}$ is bounded in $S^p[E]$ if and only if there is an analytic function $\phi$ in the vector valued Besov Space $B_p^{1/p}(E)$ such that $a_n = \hat \phi(n)$ for all $n \in \N$. In particular this condition only depends on the Banach space structure of $E$. We also show that the norm of the isomorphism $\phi \mapsto (\hat \phi(i+j))_{i,j}$ grows as $\sqrt p$ as $p \to \infty$, and compute the norm of the natural projection onto the space of Hankel matrices.