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A general extrapolation theorem for absolutely summing operators. (English) Zbl 1269.47020
Let $$X, Y, E$$ be non void-sets, $${\mathcal H}(X;Y)$$ be a non-void family of mappings from $$X$$ to $$Y$$, let $$G$$ be a Banach space space and let $$K$$ be a compact Hausdorff topological space. Let $$R: K\times E\times G\to [0,\infty)$$ and $$S: {\mathcal H}(X;Y)\times E\times G\to [0,\infty)$$ be arbitrary mappings and $$1\leq t<\infty$$. A mapping $$f\in {\mathcal H}(X;Y)$$ is called $$RS$$-abstract $$t$$-summing if there exists $$C\geq 0$$ such that $\left(\sum_{j=1}^mS(f,x_j,b_j)^t\right)^{1/t}\leq C\sup_{\varphi\in K}\left(\sum_{j=1}^m R(\varphi,x_j,b_j)^t\right)^{1/t}$ for all $$x_1,\dots, x_m\in E$$, $$b_1,\dots, b_m\in G$$ and $$m\in\mathbb N$$. Let $${\mathcal H}_{RS,t}(X;Y)=\{f\in {\mathcal H}(X;Y):f \text{ is }RS\text{-abstract}\;t\text{-summing}\}$$.
In the paper under review, the authors prove a general version of the extrapolation theorem for absolutely summing operators. Precisely, they show the following. Let $$X$$ be a topological space, $$E=X\times X$$ and $$K$$ be a compact Hausdorff space such that $$X$$ is embedded in $$C(K)$$. Let $$1<r<p<\infty$$. If $${\mathcal H}_{RS,p}(X;\ell_p)={\mathcal H}_{RS,r}(X;\ell_p)$$, then, for any Banach space $$Y$$, $${\mathcal H}_{RS,p}(X;Y)={\mathcal H}_{RS,1}(X;Y)$$.
This result extends the classical theorem due to B. Maurey and also contains the extrapolation theorem for Lipschitz $$p$$-summing operators and new extrapolation type theorems.

##### MSC:
 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 46T99 Nonlinear functional analysis
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