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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2305.13270 |
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| _version_ | 1866913985004044288 |
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| author | Gupta, Rajeev Misra, Gadadhar Ray, Samya Kumar |
| author_facet | Gupta, Rajeev Misra, Gadadhar Ray, Samya Kumar |
| contents | We investigate a Grothendieck-type inequality for pairs of Banach spaces $E,F$ assuming $E$ is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a $C >0$ such that $\|A\otimes \operatorname{id}_{F}\|_{E_m\check{\otimes}F\to E_n^*\hat{\otimes}F}\leqslant C \|A\|_{E_m\to E_n^*}$ for all $m,n\in\mathbb{N},$ where $\dim E_n=n$, then both $F$ and $F^*$ must have finite cotype. Moreover, assuming that $F$ has the bounded approximation property and that the conjecture in \cite{PisierDuality} has an affirmative answer, we show that $(E_n^*)_{n\geqslant 1}$ satisfies G.T. uniformly. We show that the Grothendieck-type constant defined for a pair of Banach spaces $(E,F)$ is closely related to another interesting quantity introduced recently in \cite{XOR games and GT} comparing the projective and injective norms on the tensor product of two finite-dimensional Banach spaces $E$ and $F$. We also study analogously the constants appearing in these extremal problems by restricting only to non-negative tensors. For contractive \emph{little} Parrott homomorphisms $\varrho_V : H^\infty(Ω) \to M_{n}$, where $Ω$ is the dual unit ball of a finite dimensional Banach space $(E,\|\cdot\|)$, we prove the sharp estimate $
\|\varrho_V\|_{\mathrm{cb}}\leq\sqrt{γ(E)}, $
$γ(E)$ being the positive Grothendieck constant associated with the pair $(E, \ell^n_2)$. %\st{with extremal cases achieving equality.} This yields a new proof of \cite[Theorem 2.1]{Davidchoi} using the lower bound $K_G^+(\ell_\infty^4,\ell_2^2) \geq 1.1658$ obtained in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_13270 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a variant of the Grothendieck inequality and estimates on tensor product norms Gupta, Rajeev Misra, Gadadhar Ray, Samya Kumar Functional Analysis We investigate a Grothendieck-type inequality for pairs of Banach spaces $E,F$ assuming $E$ is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a $C >0$ such that $\|A\otimes \operatorname{id}_{F}\|_{E_m\check{\otimes}F\to E_n^*\hat{\otimes}F}\leqslant C \|A\|_{E_m\to E_n^*}$ for all $m,n\in\mathbb{N},$ where $\dim E_n=n$, then both $F$ and $F^*$ must have finite cotype. Moreover, assuming that $F$ has the bounded approximation property and that the conjecture in \cite{PisierDuality} has an affirmative answer, we show that $(E_n^*)_{n\geqslant 1}$ satisfies G.T. uniformly. We show that the Grothendieck-type constant defined for a pair of Banach spaces $(E,F)$ is closely related to another interesting quantity introduced recently in \cite{XOR games and GT} comparing the projective and injective norms on the tensor product of two finite-dimensional Banach spaces $E$ and $F$. We also study analogously the constants appearing in these extremal problems by restricting only to non-negative tensors. For contractive \emph{little} Parrott homomorphisms $\varrho_V : H^\infty(Ω) \to M_{n}$, where $Ω$ is the dual unit ball of a finite dimensional Banach space $(E,\|\cdot\|)$, we prove the sharp estimate $ \|\varrho_V\|_{\mathrm{cb}}\leq\sqrt{γ(E)}, $ $γ(E)$ being the positive Grothendieck constant associated with the pair $(E, \ell^n_2)$. %\st{with extremal cases achieving equality.} This yields a new proof of \cite[Theorem 2.1]{Davidchoi} using the lower bound $K_G^+(\ell_\infty^4,\ell_2^2) \geq 1.1658$ obtained in this paper. |
| title | On a variant of the Grothendieck inequality and estimates on tensor product norms |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2305.13270 |