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.

Abstract extrapolation theorems for absolutely summing nonlinear operators

“…A series of recent works [6][7][8][28][29][30] on Pietsch Domination-Factorization Theorems have shown that the domination theorem actually needs almost no linear structure, and a quite general version is in fact valid (see Theorem 2.2 below). This general approach recovers several previous Pietsch-type domination theorems (see [8]) and also rapidly found applications in different contexts (see [1,11,13]). …”

“…The nonlinear theory of absolutely summing operators has been rapidly developed by several authors (see, e.g., [1,4,[6][7][8][10][11][12]18,20,[22][23][24][27][28][29][30][31]33]). A natural question is whether extrapolation theorems hold for nonlinear summing operators.…”

A general Extrapolation Theorem for absolutely summing operators

“…This general approach recovers several Pietsch Domination type theorems (see [5]) and also rapidly found applications in different contexts (see [1,8]). It is worth mentioning that the recent interesting version of the Pietsch Domination Theorem for Lipschitz (p; q; r)-summing operators proved in [7, Theorem 5.4 (a)⇒(b)] can also be obtained as a simple application of the general result from [20].…”

“…where B X # is the unit ball of the Lipschitz dual X # of X. Since the publication of Farmer and Johnson's paper, the concept of Lipschitz p-summing maps has attracted the attention of several authors (see [7][8][9]). We shall be mainly interested in the Extrapolation Theorem:…”

Nonlinear absolutely summing operators revisited

“…Since then, several authors were attracted by the subject and also non-multilinear approaches have appeared (see [16,17,37,43,45,57]). The adequate way of lifting the notion of a given operator ideal to the multilinear and polynomial settings is a delicate matter.…”