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Proceedings of the American Mathematical Society

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Lipschitz $ (q,p)$-mixing operators


Author: Javier Alejandro Chávez-Domínguez
Journal: Proc. Amer. Math. Soc. 140 (2012), 3101-3115
MSC (2010): Primary 46B28, 46T99, 47H99, 47J99, 47L20
DOI: https://doi.org/10.1090/S0002-9939-2011-11140-7
Published electronically: January 3, 2012
MathSciNet review: 2917083
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Abstract: Several useful results in the theory of $ p$-summing operators, such as Pietsch's composition theorem and Grothendieck's theorem, share a common form: for certain values $ q$ and $ p$, there is an operator such that whenever it is followed by a $ q$-summing operator, the composition is $ p$-summing. This is precisely the concept of $ (q,p)$-mixing operators, defined and studied by A. Pietsch. On the other hand, J. Farmer and W. B. Johnson recently introduced the notion of a Lipschitz $ p$-summing operator, a nonlinear generalization of $ p$-summing operators. In this paper, a corresponding nonlinear concept of Lipschitz $ (q,p)$-mixing operators is introduced, and several characterizations of it are proved. An interpolation-style theorem relating different Lipschitz $ (q,p)$-mixing constants is obtained, and it is used to show reversed inequalities between Lipschitz $ p$-summing norms.


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Additional Information

Javier Alejandro Chávez-Domínguez
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: jcdom@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11140-7
Received by editor(s): November 17, 2010
Received by editor(s) in revised form: March 21, 2011
Published electronically: January 3, 2012
Additional Notes: Partially supported by NSF grants DMS-0503688 and DMS-0852434.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 By the author