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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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