On uniformly differentiable mappings from ℿ/span>_∿/sub>(Γ)

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• 作者：Petr Há ; jek^a ; ^b ; ^{hajek@math.cas.cz" class="auth_mail" title="E-mail the corresponding author} ; Eva Pernecká ; ^c ; ^{e.pernecka@impan.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Smooth operator ; ℓ_&infin ; (Γ)ℓ&infin ; (Γ)
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 July 2016
• 年：2016
• 卷：439
• 期：1
• 页码：125-134
• 全文大小：362 K

文摘

In 1970 Haskell Rosenthal proved that if X   is a Banach space, Γ is an infinite index set, and T:ℓ_∞(Γ)→X$T:{\ell }_{\infty }(\mathrm{\Gamma })\to X$ is a bounded linear operator such that inf_γ∈Γ⁡‖T(e_γ)‖>0${\inf }_{\gamma \in \mathrm{\Gamma }}\Vert T(e_{\gamma })\Vert >0$ then T   acts as an isomorphism on abac76bb456eab2755084605736" title="Click to view the MathML source">ℓ_∞(Γ^&prime;)${\ell }_{\infty }({\mathrm{\Gamma }}^{\&<font\; color="red">prime</font>;})$, for some a8f0808dd0a5d" title="Click to view the MathML source">Γ^&prime;⊂Γ${\mathrm{\Gamma }}^{\&<font\; color="red">prime</font>;}\subset \mathrm{\Gamma }$ of the same cardinality as Γ. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of Γ, we show that if F:B_ℓ∞(Γ)→X$F:B_{{\ell }_{\infty }(\mathrm{\Gamma })}\to X$ is uniformly differentiable and such that inf_γ∈Γ⁡‖F(e_γ)−F(0)‖>0${\inf }_{\gamma \in \mathrm{\Gamma }}\Vert F(e_{\gamma })-F(0)\Vert >0$ then there exists a8ea332a3" title="Click to view the MathML source">x∈B_ℓ∞(Γ)$x\in B_{{\ell }_{\infty }(\mathrm{\Gamma })}$ such that dF(x)[⋅]$dF(x)[\cdot ]$ is a bounded linear operator which acts as an isomorphism on abac76bb456eab2755084605736" title="Click to view the MathML source">ℓ_∞(Γ^&prime;)${\ell }_{\infty }({\mathrm{\Gamma }}^{\&<font\; color="red">prime</font>;})$, for some a8f0808dd0a5d" title="Click to view the MathML source">Γ^&prime;⊂Γ${\mathrm{\Gamma }}^{\&<font\; color="red">prime</font>;}\subset \mathrm{\Gamma }$ of the same cardinality as Γ.