Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre

Details Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre

1992-10-08

arxiv.org/abs/math/9210206v1

Mathematics

Functional Analysis

Scientific paper

Let $ X=(X_0,X_1)$ and $ Y=(Y_0,Y_1)$ be Banach couples and suppose $T: X\to Y$ is a linear operator such that $T:X_0\to Y_0$ is compact. We consider the question whether the operator $T:[X_0,X_1]_{\theta}\to [Y_0,Y_1]_{\theta}$ is compact and show a positive answer under a variety of conditions. For example it suffices that $X_0$ be a UMD-space or that $X_0$ is reflexive and there is a Banach space so that $X_0=[W,X_1]_{\alpha}$ for some $0<\alpha<1.$

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