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Interpolation of compact operators in the multidimensional case. (English) Zbl 1170.46021
Monografii Matematice (Timişoara) 76. Timişoara: Universitatea de Vest din Timişoara, Facultatea de Matematică. ii, 63 p. (2003).
Theorems of Krasnosel’skii type, especially Cwikel’s result for general Banach spaces, concerning the properties of compact operators acting between Banach spaces under interpolation, are investigated in the multi-dimensional case. The objects of study are $$(n+1)$$-tuples $$\overline{A}=(A_0,A_1,\dots,A_n)$$ of Banach spaces $$A_i$$ which are continuously embedded in a Hausdorff space $$\mathcal{U},$$ and, for two such $$(n+1)$$-tuples $$\overline{A},\overline{B}$$, maps $$T:\overline{A}\rightarrow\overline{B},$$ whose restrictions to the $$A_i$$ are continuous homomorphisms into $$B_i$$, the $$i$$-th member of $$\overline{B}$$. Analogues of Peetre’s $$K$$- and $$J$$-functionals are defined, and the $$K$$ and $$J$$ real interpolation methods of Sparr, Fernandez and of Cobos-Peetre [cf. F. Cobos and J. Peetre, Proc. Lond. Math. Soc. 63, 371–400 (1991; Zbl 0702.46047)] are presented in this multi-dimensional setting. These methods are then used, in turn, to examine the validity of Cwikel’s theorem. The behaviour of the measure of non-compactness of the map $$T$$ under real interpolation is also investigated by Sparr’s methods.
##### MSC:
 46B70 Interpolation between normed linear spaces 46M35 Abstract interpolation of topological vector spaces 46B50 Compactness in Banach (or normed) spaces 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47B07 Linear operators defined by compactness properties