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

Abstract

We show that for any $1<p<\infty$, the space $Hank_p(\mathbb{R}_+)\subseteq B(L^p(\mathbb{R}_+))$ of all Hankel operators on $L^p(\mathbb{R}_+)$ is equal to the $w^*$-closure of the linear span of the operators $\theta_u\colon L^p(\mathbb{R}_+)\to L^p(\mathbb{R}_+)$ defined by $\theta_uf=f(u-\,\cdotp)$, for $u>0$. We deduce that $Hank_p(\mathbb{R}_+)$ 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 $m\colon \mathbb{R}_+^*\to \mathbb{C}$ is the symbol of a $p$-completely bounded multiplier $Hank_p(\mathbb{R}_+)\to Hank_p(\mathbb{R}_+)$ if and only if there exist $\alpha\in L^\infty(\mathbb{R}_+;L^p(\Omega))$ and $\beta\in L^\infty(\mathbb{R}_+;L^{p'}(\Omega))$ such that $m(s+t)=\langle\alpha(s),\beta(t)\rangle$ for a.e. $(s,t)\in\mathbb{R}_+^{*2}$. We also give analogues of these results in the (easier) discrete case.