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Real interpolation and closed operator ideals. (English) Zbl 1051.46013
Let \(T\) be a bounded linear operator between two Banach couples \(\overline A=(A_0,A_1)\) and \(\overline B=(B_0,B_1)\) that, as an operator from \(A_0\cap A_1\) into \(B_0+B_1\), belongs to an injective and surjective operator ideal \({\mathcal I}\). Two of the main results of the paper under review refer to the so-called \(\Sigma_\Gamma\) condition, which ensures that \(T\in{\mathcal I}(\overline A_{\Gamma,K}; \overline B_{\Gamma,K})\) (and \(T\in{\mathcal I}(\overline A_{\Gamma,J}; \overline B_{\Gamma,J})\)) for a wide class of Banach sequence lattices \(\Gamma\). Here \(\overline A_{\Gamma,K}\) and \(\overline A_{\Gamma,J}\) stand for the real K-method and J-method of interpolation with respect to \(\Gamma\). The results of the paper apply to weakly compact operators, Rosenthal operators and Banach-Saks operators.

MSC:
46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological vector spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L20 Operator ideals
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