Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$-completely bounded multipliers
Loris Arnold, Christian Le Merdy, Safoura Zadeh
TL;DR
The paper advances the theory of Hankel operators on half-lines by proving that Hank_p(R_+) coincides with the w*-closure of the span of shift-type Hankel operators and is the dual of the half-line Figa–Talamanca–Herz algebra A_p(R_+). It then provides a complete characterization of p-completely bounded multipliers on Hankel_p(R_+) via Bochner-valued factorizations: a multiplier with symbol m exists if and only if there is a measure space (Ω, μ) and α ∈ L^∞(R_+; L^p(Ω)), β ∈ L^∞(R_+; L^{p'}(Ω)) such that m(s+t) = ⟨α(s), β(t)⟩ for a.e. (s,t). The results extend the well-known p=2 theory and its discrete analogues (Hank_p(N)), establishing a unified, operator-space framework for Hankel operators and their multipliers on both continuous and discrete half-lines. This yields a robust duality and a precise multiplier criterion that generalizes ALZ and Nehari-type descriptions beyond the Hilbert space setting, with potential applications to noncommutative harmonic analysis on groups and related areas.
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 $θ_u\colon L^p(\mathbb{R}_+)\to L^p(\mathbb{R}_+)$ defined by $θ_uf=f(u-\,\cdotp)$, 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 $α\in L^\infty(\mathbb{R}_+;L^p(Ω))$ and $β\in L^\infty(\mathbb{R}_+;L^{p'}(Ω))$ such that $m(s+t)=\langleα(s),β(t)\rangle$ for a.e. $(s,t)\in\mathbb{R}_+^{*2}$. We also give analogues of these results in the (easier) discrete case.