Extension of operators defined on reflexive subspaces of \(L^1\) of \(L^1/H_0^1\). (English. Russian original) Zbl 1037.46016

Kislyakov, S. V.

J. Math. Sci., New York 115, No. 2, 2147-2156 (2003); translation from Zap. Nauchn. Semin. POMI 270, 103-123 (2000).

Author’s summary: “Interpolation theory is used to develop a general pattern for proving the extension theorems mentioned in the title. In the case where the range space \(G\) is a w\(^*\)-closed subspace of \(L^\infty\) or \(H^\infty\) with reflexive annihilator \(F\), a necessary and sufficient condition on \(G\) is found for such an extension always to be possible. Specifically, \(F\) must be Hilbertian and become complemented in \(L^p\) (\(1 < p \leq 2\)) after a suitable change of density.”
This is a very classical-looking paper (most references date from before 1992) that offers a new way to look at known extension results, and provides for them a unified view and some improvements.

Reviewer: Vania Mascioni (Muncie)

MSC:

46B25	Classical Banach spaces in the general theory
46B20	Geometry and structure of normed linear spaces
46B40	Ordered normed spaces
46B70	Interpolation between normed linear spaces

Keywords:

extension; operator; reflexive subspace; interpolation