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Real and complex interpolation and extrapolation of compact operators. (English) Zbl 0787.46062

The author settles a long standing important problem of real interpolation theory. If \(T\) is a linear operator mapping the Banach interpolation couple \((A_ 0,A_ 1)\) to the couple \((B_ 0,B_ 1)\) so that \(T: A_ 0\to B_ 0\) is continuous and \(T: A_ 1\to B_ 1\) is compact, then for \(0<\vartheta<1\) and \(q\geq 1\), \(T: (A_ 0,A_ 1)_{\vartheta,q}\to (B_ 0,B_ 1)_{\vartheta,q}\) compactly. The result for two-sided compactness, i.e. under the assumption that both \(T: A_ 0\to B_ 0\) and \(T: A_ 1\to B_ 1\) are compact goes back to Hayakawa, 1969.
Some progress on the related, important problem in the context of complex interpolation is also described.
Reviewer: Y.Sagher (Chicago)

MSC:

46M35 Abstract interpolation of topological vector spaces
47B07 Linear operators defined by compactness properties
46B70 Interpolation between normed linear spaces
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References:

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