Interpolation functors and duality
This book is a presentation of a new method of approach, due to the authors, to the duality of interpolation spaces. There are three difficulties in the creation of the duality theory for interpolation spaces. The first difficulty is that if \(\bar X=(X_ 0,X_ 1)\) is a Banach couple, then the dual spaces \((X_ 0',X_ 1')\) need not be a dual couple. Even if \((X_ 0',X_ 1')\) is a Banach couple we cannot speak in general about bidual. The second difficulty is that if X is an interpolation space for \(\bar X,\) then, even if \(\bar X'\) is a Banach couple, the dual space X' need not be an intermediate space for \(\bar X\) (and interpolation space, of course). The third difficulty is that there is no general rule of construction of a dual method of interpolation for a given method. The authors show that if we work with some category of doolittle diagrams, which is larger than the category of Banach couples, then the duality theory may be obtained. Though this categorical language is rather hard for analysts it leads to interesting results both in the general setting and applications to concrete methods of interpolation. The authors give the general definition of dual functor and a related, very useful notion of computable functor. Note that previous works of the authors on this topic lead to the appearance of the series of works of the Soviet mathematician \textit{M. Aizenshtein}, pupil of Yu. Brudryi [see: Investigations in the theory of functions of several real variables. Yaroslavl', 3-10; 11-35 (1986); Geometrical questions of the theory of functions and sets, Kalinin, 6-16 (1986)].
scientific article; zbMATH DE number 3992206