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ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

Part of: Methods of category theory in functional analysis Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals

Published online by Cambridge University Press:  20 August 2020

Jesús M. F. Castillo Open the ORCID record for Jesús M. F. Castillo [Opens in a new window] ,
Willian H. G. Corrêa Open the ORCID record for Willian H. G. Corrêa [Opens in a new window] ,
Valentin Ferenczi Open the ORCID record for Valentin Ferenczi [Opens in a new window]  and
Manuel González Open the ORCID record for Manuel González [Opens in a new window]
Jesús M. F. Castillo
Affiliation:
Instituto de Matemáticas, Universidad de Extremadura, Avenida de Elvas s/n, 06011Badajoz, Spain (castillo@unex.es)
Willian H. G. Corrêa
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090São Paulo SP, Brazil (willhans@ime.usp.br; ferenczi@ime.usp.br)
Valentin Ferenczi
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090São Paulo SP, Brazil (willhans@ime.usp.br; ferenczi@ime.usp.br)
Manuel González
Affiliation:
Departamento de Matemáticas, Universidad de Cantabria, Avenida de los Castros s/n, 39071Santander, Spain (manuel.gonzalez@unican.es)
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Abstract

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

Keywords

stability of splittingtwisted sums of Banach spacescomplex interpolation

MSC classification

Primary: 46B70: Interpolation between normed linear spaces
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46M18: Homological methods (exact sequences, right inverses, lifting, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 1 , January 2022 , pp. 303 - 334
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

The research of the first author was supported in part by Project IB16056 de la Junta de Extremadura; the research of the first and fourth authors was supported in part by Project MTM2016-76958, Spain. The research of the second author was supported in part by CNPq, grant 140413/2016-2, CAPES, PDSE program 88881.134107/2016-0, and FAPESP, grants 2016/25574-8 and 2018/03765-1. The research of the third author was supported by FAPESP, grants 2013/11390-4, 2015/17216-1, 2016/25574-8 and by CNPq, grants 303034/2015-7 and 303731/2019-2.

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