A general extrapolation theorem for absolutely summing operators. (English) Zbl 1269.47020

Pellegrino, D.; Santos, J.; Seoane-Sepúlveda, J. B.

Bull. Lond. Math. Soc. 44, No. 6, 1292-1302 (2012).

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.

Reviewer: Angela Albanese (Lecce)

MSC:

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

Keywords:

\(RS\)-abstract \(t\)-summing mapping; extrapolation theorem