Lipschitz $(q,p)$-mixing operators

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.