an:05485696
Zbl 1170.46021
Cofan, Nicolae; Stan, Ilie
Interpolation of compact operators in the multidimensional case
EN
Monografii Matematice (Timi??oara) 76. Timi??oara: Universitatea de Vest din Timi??oara, Facultatea de Matematic??. ii, 63~p. (2003).
2003
b
46B70 46M35 46B50 47H09 47B07
real interpolation; Peetre's K- and J-functionals; Krasnosel'skij-type theorems; compact operator; measure of noncompactness
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.\ \textit{F.\,Cobos} and \textit{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.
W. D. Evans (Cardiff)
Zbl 0702.46047