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Interpolation of spectrum of bounded operators on Lebesgue spaces. (English) Zbl 0738.47001
Summary: Let \(\mu\) be a \(\sigma\)-finite positive measure. Assume \(1\leq p<s<\infty\). Let \(T\) be a linear operator on \(L^ p(\mu)\cap L^ s(\mu)\) that has bounded extensions \(T_ p\) and \(T_ s\) on \(L^ p(\mu)\) and \(L^ s(\mu)\) respectively. Then \(T\) has a bounded extension \(T_ r\) on \(L^ r(\mu)\), \(p\leq r\leq<s\). The aim of this paper is to study the relationship between the spectral and Fredholm properties of the operator \(T_ r\) and those of \(T_ p\) and \(T_ s\).

MSC:
47A10 Spectrum, resolvent
47A53 (Semi-) Fredholm operators; index theories
46B70 Interpolation between normed linear spaces
47C05 Linear operators in algebras
46M35 Abstract interpolation of topological vector spaces
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