Duality for Lipschitz p-summing operators

Abstract

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banachspace-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach spaceanswering a question of J. Farmer and W.B. Johnson (2009) [6] -and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J. T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch's (p, r, s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q, p)-dominated operators.

Lipschitz compact operators

“…Moreover, Chen and Zheng [5] introduced and studied strongly Lipschitz p-nuclear operators and Lipschitz p-nuclear operators. Chávez-Domínguez introduced and investigated Lipschitz (p, r, s)-summing operators and Lipschitz (q, p)-mixing operators in [2] and [3], respectively.…”

On the composition ideals of Lipschitz mappings

“…In [4], the author has studied the class Π L p (X; E) of Lipschitz p-summing operators. He has defined a norm on the space of molecules F (X; E) of which we have the next duality Π L p (X; E * ) = F csp (X; E) * , where cs p is defined by…”

Abstract extrapolation theorems for absolutely summing nonlinear operators

“…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.…”