Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T12:20:42.086Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Relationship between Interpolation of Banach Algebras and Interpolation of Bilinear Operators

Published online by Cambridge University Press:  20 November 2018

Fernando Cobos  and
Luz M. Fernández-Cabrera
Fernando Cobos
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain e-mail: cobos@mat.ucm.es
Luz M. Fernández-Cabrera
Affiliation:
Sección Departamental de Matemática Aplicada, Escuela de Estadística, Universidad Complutense de Madrid, 28040 Madrid, Spain e-mail: luz fernandez-c@mat.ucm.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if the general real method ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$.

Keywords

46B7046M3546H05real interpolationbilinear operatorsBanach algebras
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 1 , 01 March 2010 , pp. 51 - 57
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Bergh, J. and Löfström, J., Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin-New York, 1976.Google Scholar
[2] Bishop, E. A., Holomorphic completion, analytic continuation, and the interpolation of semi-norms. Ann. of Math. 78(1963), 468–500. doi:10.2307/1970537Google Scholar
[3] Blanco, A., Kaijser, S., and Ransford, T. J., Real interpolation of Banach algebras and factorization of weakly compact homomorphisms. J. Funct. Anal. 217(2004), no. 1, 126–141. doi:10.1016/j.jfa.2004.03.011Google Scholar
[4] Brudnyĭ, Y. and Krugljak, N., Interpolation functors and interpolation spaces. Vol. 1. North-Holland Mathematical Library 47, North-Holland Publishing Co., Amsterdam, 1991.Google Scholar
[5] Calder ón, A.-P., Intermediate spaces and interpolation, the complex method. Studia Math. 24(1964), 113–190.Google Scholar
[6] Cobos, F., Fernández-Cabrera, L. M., and Martínez, A., Compact operators between K- and J-spaces. Studia Math. 166(2005), no. 3, 199–220. doi:10.4064/sm166-3-1Google Scholar
[7] Cobos, F., Fernández-Cabrera, L. M., and Martínez, A., On interpolation of Banach algebras and factorization of weakly compact homomorphisms. Bull. Sci. Math. 130(2006), no. 7, 637–645. doi:10.1016/j.bulsci.2005.12.003Google Scholar
[8] Cobos, F., Fernández-Cabrera, L. M., and Martínez, A., Abstract K and J spaces and measure of non-compactness. Math. Nachr. 280(2007), no. 15, 1698–1708. doi:10.1002/mana.200510572Google Scholar
[9] Kaijser, S., Interpolation of Banach algebras and open sets, Integr. Equ. Oper. Theory 41(2001) 189–222. doi:10.1007/BF01295305Google Scholar
[10] Nilsson, P., Reiteration theorems for real interpolation and approximation spaces. Ann. Mat. Pura Appl. 132(1982), 291–330. doi:10.1007/BF01760986Google Scholar
[11] Nilsson, P., Interpolation of Calderón and Ovčhinnikov pairs. Ann. Mat. Pura Appl. 134(1983), 201–232. doi:10.1007/BF01773505Google Scholar
[12] Peetre, J., A theory of interpolation of normed spaces. Notas de Matemática 39, Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, 1968.Google Scholar
[13] Triebel, H., Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library 18, North-Holland Publishing Co., Amsterdam-New York, 1978.Google Scholar
[14] Zafran, M., The dichotomy problem for homogeneous Banach algebras. Ann. of Math. 108(1978), no. 1, 97–105. doi:10.2307/1970931Google Scholar