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On Sparr and Fernandez’s interpolation methods of Banach spaces. (English) Zbl 1033.46058

Interpolation spaces as defined in [G. Sparr, Ann. Math. Pura Appl., IV Ser. 99, 247–316 (1974; Zbl 0282.46002) and D. L. Fernandez, Stud. Math. 65, 175–201 (1979; Zbl 0462.46051)] are investigated for 4-tuples of Banach spaces \(\overline{A}=(A_0,A_1,A_2,A_3)\), where \(A_i\) is of the class \(\mathcal C(\theta_i,X,Y)\), \(0<\theta_i<1\). It is well-known that, in contrast to the case of Banach couples, the so-called equivalence theorem fails, in general, for Banach \(n\)-tuples. Here, it is shown that if \(\theta_i\) are suitably chosen then the \(J\)- and \(K\)-methods of interpolation coincide and are equal to the space \((X,Y)_{\eta,p}\) for certain \(\eta\), \(p\). Some applications to \(L^p\)-spaces, to semigroups of operators and to compact operators are given.

MSC:

46M35 Abstract interpolation of topological vector spaces
46B70 Interpolation between normed linear spaces
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