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Remarks on interpolation properties of Schatten classes. (English) Zbl 0821.46023

Let \(H_ 1\subset H_ 0\), \(K_ 1\subset K_ 0\) be Hilbert spaces with \(H_ 1\) dense in \(H_ 0\), \(K_ 1\) dense in \(K_ 0\) and the embeddings \(H_ 1\hookrightarrow H_ 0\), \(K_ 1\hookrightarrow K_ 0\) compact. Denote by \(\{H_ \theta\}_{0\leq \theta\leq 1}\) (resp. \(\{K_ \theta\}_{0\leq \theta\leq 1}\)) the Hilbert scale obtained from \((H_ 0, H_ 1)\) (resp. \((K_ 0, K_ 1)\)) by complex interpolation. Let \(T\) be a linear operator whose restrictions \(T: H_ 0\to K_ 0\) and \(T: H_ 1\to K_ 1\) are bounded. It was supposed in the literature that nuclearity of \(T: H_ 0\to K_ 0\) is enough to guarantee that the interpolated operator \(T: H_ \theta\to K_ \theta\) is also nuclear for any \(0< \theta< 1\).
Nevertheless, this is not true in general as we show in this paper. Our counterexample refers not only to nuclear operators but to any other Schatten \(p\)-class \((1\leq p< \infty)\). We also establish that the following complex interpolation formula for Hilbert-Schmidt operators \[ [S_ 2(H_ 0, K_ 0), S_ 2(H_ 1, K_ 1)]_ \theta= S_ 2(H_ \theta, K_ \theta) \] holds with equal norms.
Reviewer: F.Cobos (Madrid)

MSC:

46B70 Interpolation between normed linear spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46M35 Abstract interpolation of topological vector spaces
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