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ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

Part of: Normed linear spaces and Banach spaces; Banach lattices Methods of category theory in functional analysis 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.

References

Alvarez, J. A., Alvarez, T. and González, M., The gap between subspaces and perturbation of non-semi-Fredholm operators, Bull. Aust. Math. Soc. 45 (1992), 369–376.CrossRefGoogle Scholar
Avilés, A., Cabello Sánchez, F., Castillo, J. M. F., González, M. and Moreno, Y., Separably Injective Banach Spaces, Lecture Notes in Mathematics, Volume 2132 (Springer, Berlin, Heidelberg, New-York, 2016).CrossRefGoogle Scholar
Bak, J. and Newman, D. J., Complex Analysis (Springer, Berlin, Heidelberg, New-York, 1982).Google Scholar
Bergh, J. and Löfström, J., Interpolation Spaces. An Introduction (Springer, Berlin, Heidelberg, New-York, 1976).CrossRefGoogle Scholar
Cabello, F., Castillo, J. M. F. and Kalton, N. J., Complex interpolation and twisted twisted Hilbert spaces, Pacific J. Math. 276 (2015), 287–307.CrossRefGoogle Scholar
Cabello, F., Castillo, J. M. F. and Suárez, J., On strictly singular nonlinear centralizers, Nonlinear Anal. 75 (2012), 3313–3321.Google Scholar
Calderón, A.-P., Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.CrossRefGoogle Scholar
Casazza, P. G. and Kalton, N. J., Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces, Israel J. Math. 95 (1996), 349–373.CrossRefGoogle Scholar
Castillo, J. M. F., Ferenczi, V. and González, M., Singular twisted sums generated by complex interpolation, Trans. Amer. Math. Soc. 369 (2017), 4671–4708.CrossRefGoogle Scholar
Castillo, J. M. F. and González, M., Three-space Problems in Banach Space Theory, Lecture Notes in Mathematics, Volume 1667 (Springer, Berlin, Heidelberg, 1997).CrossRefGoogle Scholar
Chaatit, F., On uniform homeomorphisms of the unit spheres of certain Banach lattices, Pacific J. Math. 168 (1995), 11–31.CrossRefGoogle Scholar
Coifman, R. R., Cwikel, M., Rochberg, R., Sagher, Y. and Weiss, G., A theory of complex interpolation for families of Banach spaces, Adv. Math. 43 (1982), 203–229.CrossRefGoogle Scholar
Corrêa, W. H. G., Type, cotype and twisted sums induced by complex interpolation, J. Funct. Anal. 274 (2018), 797–825.CrossRefGoogle Scholar
Cwikel, M., Jawerth, B., Milman, M. and Rochberg, R., Differential estimates and commutators in interpolation theory, in Analysis at Urbana II, (ed. Berkson, E. R., Peck, N. T. and Uhl, J.), London Mathematical Society Lecture Note Series, Volume 138, pp. 170–220 (Cambridge University Press, Cambridge, 1989).CrossRefGoogle Scholar
Cwikel, M. and Janson, S., Real and complex interpolation methods for finite and infinite families of Banach spaces, Adv. Math. 66(3) (1987), 234–290.CrossRefGoogle Scholar
Daher, M., Homéomorphismes uniformes entre les sphères unité des espaces d’interpolation, Canad. Math. Bull. 38 (1995), 286–294.CrossRefGoogle Scholar
Domanski, P. and Mastyło, M., Characterization of splitting for Fréchet-Hilbert spaces via interpolation, Math. Ann. 339 (2007), 317–340.CrossRefGoogle Scholar
Duren, P. L., Theory of H p Spaces (Academic Press, New York, 1970).Google Scholar
Ferenczi, V., A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199–225.CrossRefGoogle Scholar
Hernández, E., Intermediate spaces and the complex method of interpolation for families of Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 13 (1986), 245–266.Google Scholar
Kalton, N. J., Nonlinear commutators in interpolation theory, Mem. Amer. Math. Soc. 385 (1988).Google Scholar
Kalton, N. J., Differentials of complex interpolation processes for Köthe function spaces, Trans. Amer. Math. Soc. 333 (1992), 479–529.Google Scholar
Kalton, N. J., Hermitian operators on complex Banach lattices and a problem of Garth Dales, J. Lond. Math. Soc. (2) 86 (2012), 641–656.CrossRefGoogle Scholar
Kalton, N. J. and Montgomery-Smith, S., Interpolation of Banach Spaces, (ed. Johnson, W. B. and Lindenstrauss, J.), Chapter 36 in Handbook of the Geometry of Banach Spaces, Volume 2, pp. 1131–1175 (North-Holland, The Netherlands, 2003).Google Scholar
Kalton, N. J. and Ostrovskii, M., Distances between Banach spaces, Forum Math. 11 (1999), 17–48.CrossRefGoogle Scholar
Laursen, K. B. and Neumann, M. M., An Introduction to Local Spectral Theory, London Mathematical Society Monographs New Series, Volume 20 (The Clarendon Press, Oxford University Press, New York, 2000).Google Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I. Sequence Spaces (Springer, Berlin, Heidelberg, New-York, 1977).Google Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces II. Function Spaces (Springer, Berlin, Heidelberg, New-York, 1979).CrossRefGoogle Scholar
Lozanovskii, G. Y., On some Banach lattices, Sib. Math. J. 10 (1969), 419–430.CrossRefGoogle Scholar
Pisier, G., Some applications of the complex interpolation method to Banach lattices, J. Anal. Math. 35 (1979), 264–281.CrossRefGoogle Scholar
Qiu, Y., On the effect of rearrangement on complex interpolation for families of Banach spaces, Rev. Mat. Iberoam. 31 (2015), 439–460.CrossRefGoogle Scholar
Rochberg, R., Function theoretic results for complex interpolation families of Banach spaces, Trans. Amer. Math. Soc. 284 (1984), 745–758.CrossRefGoogle Scholar
Rochberg, R., Higher order estimates in complex interpolation theory, Pacific J. Math. 174 (1996), 247–267.CrossRefGoogle Scholar
Rochberg, R. and Weiss, G., Derivatives of analytic families of Banach spaces, Ann. of Math. (2) 118 (1983), 315–347.CrossRefGoogle Scholar
Stafney, J., Analytic interpolation of certain multiplier spaces, Pacific J. Math 32 (1970), 241–248.CrossRefGoogle Scholar
Zafran, M., Spectral theory and interpolation of operators, J. Funct. Anal. 36 (1980), 185–204.CrossRefGoogle Scholar