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