Differentials of complex interpolation processes for Köthe function spaces

Kalton, N. J.

doi:10.1090/S0002-9947-1992-1081938-1

Differentials of complex interpolation processes for Köthe function spaces
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Trans. Amer. Math. Soc. 333 (1992), 479-529 Request permission

Abstract:

We continue the study of centralizers on Köthe function spaces and the commutator estimates they generate (see [29]). 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 $({\text {UMD}})$-space.

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