On the uniqueness of the solution of the inverse exact interpolation problem (Q1316822)

In the present paper we study the following problem [\textit{Yu. A. Brudnyi}, \textit{S. G. Krein} and \textit{E. M. Semenov}, in: Itogi Nauki Tekh., Ser. Mat. Anal. 24, 3-164 (1986; Zbl 0667.46046)]: 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 [loc. cit.], and a theorem by \textit{V. G. Zobina} [Interpolation of operators in spaces with prescribed symmetries [in Russian], Ph. D. Thesis, Kazan' State University, Kazan' (1979)]\ asserting that the couple of finite-dimensional spaces \((\ell_1^n, \ell^n_\infty)\) is uniquely determined by the collection of its exact interpolation spaces.