Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$-completely bounded multipliers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929697368047616 |
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| author | Arnold, Loris Merdy, Christian Le Zadeh, Safoura |
| author_facet | Arnold, Loris Merdy, Christian Le Zadeh, Safoura |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09481 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$-completely bounded multipliers Arnold, Loris Merdy, Christian Le Zadeh, Safoura Functional Analysis 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. |
| title | Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$-completely bounded multipliers |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2301.09481 |