An extension of the Aronszajn-Gagliardo theorem

Abstract

In 1965 Aronszajn and Gagliardo [l] proved that any interpolation space of a given Banach couple could be realized as the value of a minimal or maximal interpolation functor on the category of all Banach couples. This result was, paradoxically, both considered a basic result of the theory and ignored. It is only recently that attention has once again been focussed on it in view of the discovery by Brudnyi-Krugljak [3] and Janson [8] that there exists a strong connection between the important methods of interpolation and the Aronszajn-Gagliardo theorem. For example, Janson has shown that many interpolation functors, including the real and complex methods, are minimal or maximal extensions from a single Banach couple, hence, ‘Aronszajn-Gagliardo functors’. While the category of Banach spaces has been and continues to be the most studied setting for interpolation theory, applications have indicated the desirability and need to have a theory of interpolation in other settings. A particular need exists for the category of quasi-normed spaces, the setting of the classical Marcinkiewicz interpolation theorem of 1939 [2], in order to obtain a full generalization of this theorem. Accordingly, interpolation methods have been studied in this category by KrCe [ 131, Holmstedt [7], and Sagher [ 171, and real methods of interpolation have been defined there. Further prompted by applications to approximation theory, Peetre and Sparr [16] have developed a theory of interpolation for quasi-normed abelian groups and normed abelian groups. Categories of weaker linear structures,