Interpolation with real methods of special Banach triples. (English) Zbl 0921.46017

Mihailov, D.; Stan, L.

Proceedings of the sixth symposium of mathematics and its applications, Timişoara, Romania, November 3–4, 1995. Timişoara: Editura Mirton, 109-114 (1995).

The authors extend some results of interpolation theory for couples of Banach spaces to triples of Banach spaces. The precise setting is the following: Let \(\overline{X}= (X_1,X_2,X_3)\) be a triple of Banach spaces. Associated with \(\overline{X}\) are the Banach spaces \(\Delta\overline{X}= X_0\cap X_1\cap X_2\) and \(\sum \overline{X}= X_0+ X_1+ X_2\) endowed with the norms \(\| x\|_{\Delta\overline{X}}= \max\{\| x\|_{X_0},\| x\|_{X_1}+\| x\|_{X_2}\}\) and \[ \| x\|_{\sum\overline{X}}= \inf\{\| x_0\|_{X_0}+\| x_1\|_{X_1}+\| x_2\|_{X_2}: x= x_0+ x_1+ x_2,\;x_j\in X_j\}. \] A Banach space \(X\) is called an interpolation space with respect to \(\overline{X}\) if it is an intermediate space, that is if \(\Delta\overline{X}\) is continuously embedded in \(X\) and if \(X\) is continuously embedded in \(\sum\overline{X}\), and if any bounded linear operator \(T:\overline{X}\to \overline{X}\) is bounded from \(X\) into \(X\). Several other notions as orbits and coorbits of intermediate spaces as well as orbitally equivalence of two elements \(x\in \sum\overline{X}\) and \(y\in \sum\overline{Y}\) are studied.
Not all the proofs for the stated theorems and corollaries are given.
For the entire collection see [Zbl 0903.00031].

Reviewer: R.Mortini (Metz)

MSC:

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

Keywords:

interpolation theory; triples of Banach spaces; intermediate space; orbits; coorbits