Lipschitz \((q,p)\)-mixing operators (Q2845735)

Let \(1 \leq p,q \leq \infty\) and determine \(r\) by \(1/p = 1/q + 1/r\). \textit{A. Pietsch} [Stud. Math. 28, 333--353 (1967; Zbl 0156.37903)] introduced (linear) \((q,p)\)-mixing operators as linear operators that map weakly \(p\)-summable sequences into a \((q,p)\)-mixed sequence, i.e., one that can be expressed as the pointwise product of a weakly \(q\)-summable sequence and an \(r\)-summable scalar sequence. A~\((q,p)\)-mixing operator \(T\) has the property that \(S \circ T\) is \(p\)-summing whenever \(S\) is \(q\)-summing.NEWLINENEWLINERecently, \textit{J. D. Farmer} and \textit{W. B. Johnson} [Proc. Am. Math. Soc. 137, No. 9, 2989--2995 (2009; Zbl 1183.46020)] introduced the class of Lipschitz \(p\)-summing operators between metric spaces \(X\) and \(Y\), denoted by \(\Pi^L_p(X,Y)\) with norm \(\pi_p^L(\cdot)\).NEWLINENEWLINEIn this paper, the author introduces and studies Lipschitz \((q,p)\)-mixing operators. A map \(T: X \to Y\) is a Lipschitz \((q,p)\)-mixing operator with constant \(K\) if, for any metric space \(Z\) and any Lipschitz \(q\)-summing operator \(S: Y \to Z\), the composition \(S \circ T\) is a Lipschitz \(p\)-summing operator and \(\pi_p^L(S \circ T) \leq K \pi_q^L(S)\). The smallest such \(K\) is the mixing norm of \(T\). Several characterizations of such operators are proved both in the spirit of the Pietsch domination theorem and using the new concept of Lipschitz mixing sequences.NEWLINENEWLINEThe author also connects Lipschitz \((q,p)\)-mixing operators to operators on the space of \(E\)-valued molecules \(\mathcal{M}(X,E)\), \(E\) a Banach space, with the Chevet-Saphar norms introduced by \textit{J. A. Chávez-Domínguez} [J. Funct. Anal. 261, No. 2, 387--407 (2011; Zbl 1234.46009)].NEWLINENEWLINEThe paper concludes with several applications, including an interpolation-style theorem relating different Lipschitz \((q,p)\)-mixing constants.