Cwikel, Michael Real and complex interpolation and extrapolation of compact operators. (English) Zbl 0787.46062 Duke Math. J. 65, No. 2, 333-343 (1992). 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) Cited in 4 ReviewsCited in 44 Documents MSC: 46M35 Abstract interpolation of topological vector spaces 47B07 Linear operators defined by compactness properties 46B70 Interpolation between normed linear spaces Keywords:real interpolation; interpolation couple; two-sided compactness; complex interpolation PDF BibTeX XML Cite \textit{M. Cwikel}, Duke Math. J. 65, No. 2, 333--343 (1992; Zbl 0787.46062) Full Text: DOI References: [1] J. Bergh and J. Löfström, Interpolation spaces. An introduction , Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin, 1976. · Zbl 0344.46071 [2] A.-P. Calderón, Intermediate spaces and interpolation, the complex method , Studia Math. 24 (1964), 113-190. · Zbl 0204.13703 [3] F. Cobos, T. Kühn, and T. Schonbek, One-sided compactness results for Aronszajn-Gagliardo functors , Univ. Autonoma de Madrid Prepublicaciones. · Zbl 0787.46061 [4] F. Cobos and J. Peetre, Interpolation of compactness using Aronszajn-Gagliardo functors , Israel J. Math. 68 (1989), no. 2, 220-240. · Zbl 0716.46054 [5] K. Hayakawa, Interpolation by the real method preserves compactness of operators , J. Math. Soc. Japan 21 (1969), 189-199. · Zbl 0181.13703 [6] S. Janson, Minimal and maximal methods of interpolation , J. Funct. Anal. 44 (1981), no. 1, 50-73. · Zbl 0492.46059 [7] S. Janson, private communication . · Zbl 0839.05074 [8] M. A. Krasnosel’skiĭ, On a theorem of M. Riesz , Soviet Math. Dokl. 1 (1960), 229-231. · Zbl 0097.10202 [9] J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation , Inst. Hautes Études Sci. Publ. Math. (1964), no. 19, 5-68. · Zbl 0148.11403 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.