Interpolation of compact operators by the complex method and equicontinuity
The author investigates the relation between interpolation by the complex method of compactness operators and equicontinuity. Given a Banach couple \((A_0,A_1)\), let \({\mathcal F}(A_0, A_1)\) be the Banach space of analytic functions used in the definition of the complex method. A bounded set \(E\subset{\mathcal F}(A_0, A_1)\) is said to be \(\theta\)-equicontinuous if, for every \(\varepsilon> 0\), there is \(\delta> 0\) such that \(\|f(\theta+ it)- f(\theta+ it')\|_{[\theta]}< \varepsilon\) for all \(f\in E\), whenever \(t,t'\in R\) and \(|t-t'|< \delta\).NEWLINENEWLINENEWLINEGiven two Banach couples \((A_0, A_1)\) and \((B_0, B_1)\), the author shows that if \(T: A_0\to B_0\) is compact, then compactness of \(T:(A_0, A_1)_{[\theta]}\to (B_0, B_1)_{[\theta]}\) is equivalent to \(\theta\)-equicontinuity of \(T(B_{F(A_0, A_1)})\). Where \(B_{F(A_0, A_1)}\) is the unit ball of \({\mathcal F}(A_0, A_1)\).
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