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

Loris Arnold, Christian Le Merdy and Safoura Zadeh

Vol. 328 (2024), No. 2, 193–216

DOI: 10.2140/pjm.2024.328.193

Abstract

We show that for any $1<p<\infty$, the space ${\mathrm{Hank}<!--FUNCTION\; APPLICATION-->}_p({ℝ}_{+})\subseteq B(L^p({ℝ}_{+}))$ of all Hankel operators on $L^p({ℝ}_{+})$ is equal to the $w^{\ast }$-closure of the linear span of the operators ${𝜃}_u:L^p({ℝ}_{+})\to L^p({ℝ}_{+})$ defined by ${𝜃}_{u}f=f(u-\cdot )$, for $u>0$. We deduce that ${\mathrm{Hank}<!--FUNCTION\; APPLICATION-->}_p({ℝ}_{+})$ is the dual space of $A_p({ℝ}_{+})$, a half-line analogue of the Figà-Talamanca–Herz algebra $A_p(ℝ)$. Then we show that a function $m:{ℝ}_{+}^{\ast }\to ℂ$ is the symbol of a $p$-completely bounded multiplier ${\mathrm{Hank}<!--FUNCTION\; APPLICATION-->}_p({ℝ}_{+})\to {\mathrm{Hank}<!--FUNCTION\; APPLICATION-->}_p({ℝ}_{+})$ if and only if there exist $\alpha \in L^{\infty }({ℝ}_{+};L^p(\mathrm{\Omega }))$ and $\beta \in L^{\infty }({ℝ}_{+};L^{p^{\prime }}(\mathrm{\Omega }))$ such that $m(s+t)=⟨\alpha (s),\beta (t)⟩$ for a.e. $(s,t)\in {ℝ}_{+}^{\ast 2}$. We also give analogues of these results in the (easier) discrete case.

Keywords

$p$-complete boundedness, multipliers, Hankel operators

Mathematical Subject Classification

Primary: 47B35

Secondary: 46L07

Milestones

Received: 3 February 2023

Accepted: 12 April 2024

Published: 30 April 2024