ON THE UNIQUENESS OF THE SOLUTION OF THE INVERSE EXACT INTERPOLATION PROBLEM

In the present paper we study the following problem [1]: is a Banach couple uniquely determined by the collection of all interpolation spaces generated by it? The authors are familiar with only two results concerning this question. There are a rather special result due to Aronszajn and Gagliardo, cited in the survey [1], and a theorem by V. G. Zobina [4] asserting that the couple of finite-dimensional spaces (l1n, l(infinity)n) is uniquely determined by the collection of its exact interpolation spaces. We say that two pairs (X1, X2) and (Y1, Y2) of normed spaces are homothetic if, possibly after interchanging Y1 with Y2, X(i) coincides with Y(i) as a linear space and the norm in X(i) is a multiple of the norm in Y(i), i = 1, 2. The same notation will be used for the norm in a normed space and the corresponding norm for linear operators in this space.