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Differentials of complex interpolation processes for Köthe function spaces. (English) Zbl 0776.46033

Summary: We continue the study of centralizers on Köthe function spaces and the commutator estimates they generate [see the author, Nonlinear commutators in interpolation theory, Mem. Am. Math. Soc. 385, 85 p. (1988; Zbl 0658.46059)]. Our main result is that if \(X\) is a super-reflexive Köthe function space then for every real centralizer \(\Omega\) on \(X\) there is a complex interpolation scale of Köthe function spaces through \(X\) inducing \(\Omega\) as a derivative, up to equivalence and a scalar multiple. Thus, in a loose sense, all real centralizers can be identified with derivatives of complex interpolation processes. We apply our ideas in an appendix to show, for example, that there is a twisted sum of two Hilbert spaces which fails to be a (UMD)-space.

MSC:

46M35 Abstract interpolation of topological vector spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B70 Interpolation between normed linear spaces
47B38 Linear operators on function spaces (general)
42A50 Conjugate functions, conjugate series, singular integrals
47L30 Abstract operator algebras on Hilbert spaces
46B42 Banach lattices

Citations:

Zbl 0658.46059
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