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Spectral continuity in complex interpolation. (English) Zbl 0708.46060

Summary: Let [\({\mathcal X}_ 0,{\mathcal X}_ 1]\) be an interpolation pair in the sense of A. P. Calderón, and let A be a linear map from \({\mathcal X}_ 0+{\mathcal X}_ 1\) into itself such that A\({\mathcal X}_ j\subset {\mathcal X}_ j\) and \(A| {\mathcal X}_ j\) is a bounded map for \(j=0,1\). For \(0<t<1\), let \(A_ t\) be the restriction of A to the interpolation space \({\mathcal X}_ t\) (obtained by Calderón complex method), and let \(\sigma (A_ t)\) denote the spectrum of this operator. It is shown that the mapping \(t\to \sigma (A_ t){\hat{\;}}\) is continuous at every point \(t_ 0\in (0,1)\) such that \(\sigma (A_{t_ 0})\) has empty interior. The analogous result is true for the essential spectrum. All these results suggest that \(t\to \sigma (A_ t)\) (t\(\in (0,1))\) is always a continuous mapping. Some examples illustrate the difficulties involved in the solution of this problem.

MSC:

46M35 Abstract interpolation of topological vector spaces
46B70 Interpolation between normed linear spaces
47A10 Spectrum, resolvent
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