Hankel operators on $L^p({\mathbb{R}}_{+})$ and their $p$-completely bounded multipliers

Abstract

We show that for any $1<p<\infty$, the space $Han{k}_p({\mathbb{R}}_{+})\subseteq B(L^p({\mathbb{R}}_{+}))$ of all Hankel operators on $L^p({\mathbb{R}}_{+})$ is equal to the $w^{\ast }$-closure of the linear span of the operators $\hat{I}{¸}_u:L^p({\mathbb{R}}_{+})\to L^p({\mathbb{R}}_{+})$ defined by $\hat{I}{¸}_{u}f=f(u-\cdot )$, for $u>0$. We deduce that $Han{k}_p({\mathbb{R}}_{+})$ is the dual space of$A_p({\mathbb{R}}_{+})$, a half-line analogue of the Figa-Talamenca-Herz algebra $A_p(\mathbb{R})$. Then we show that a function $m:{\mathbb{R}}_{+}^{\ast }\to \mathbb{C}$ is the symbol of a $p$-completely bounded multiplier $Han{k}_p({\mathbb{R}}_{+})\to Han{k}_p({\mathbb{R}}_{+})$ if and only if there exist $\hat{I}\pm \in L^{\infty }({\mathbb{R}}_{+};L^p(\hat{I}\text{c◯}))$ and ${\hat{I}}^2\in L^{\infty }({\mathbb{R}}_{+};L^{p^{\prime }}(\hat{I}\text{c◯}))$ such that $m(s+t)=⟨\hat{I}\pm (s),{\hat{I}}^2(t)⟩$ for a.e. $(s,t)\in {\mathbb{R}}_{+}^{\ast 2}$. We also give analogues of these results in the (easier) discrete case.

Revises version, published in Pacific Journal of Mathematics