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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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