Lipschitz $p$-summing operators

Farmer, Jeffrey D.; Johnson, William B.

doi:10.1090/s0002-9939-09-09865-7

Cited by 71 publications

(100 citation statements)

References 9 publications

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“…Recently, Lipschitz versions of different types of bounded linear operators have been investigated by various authors. Farmer and Johnson [7] introduced the notion of Lipschitz p-summing operators and the notion of Lipschitz p-integral operators between metric spaces and proved a nonlinear version of the Pietsch factorization theorem. The Farmer-Johnson factorization theorem was used by Chen and Zheng in [4] to give a nonlinear version of Maurey's extrapolation theorem and deduce a nonlinear form of the Grothendieck's theorem.…”

Section: Introductionmentioning

confidence: 99%

Lipschitz compact operators

Jiménez-Vargas

Sepulcre

Villegas-Vallecillos

2014

Journal of Mathematical Analysis and Applications

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Abstract. We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.

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Section: Introductionmentioning

confidence: 99%

Lipschitz compact operators

Jiménez-Vargas

Sepulcre

Villegas-Vallecillos

2014

Journal of Mathematical Analysis and Applications

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“…This is a true generalization of the concept of linear p-summing operator, since it is shown in [6, Theorem 2] that the Lipschitz p-summing norm of a linear operator is the same as its p-summing norm. For the sequel, it will be useful to note that the above definition is the same if we restrict to λ j = 1 (see [6] for the proof).…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…In order to answer Question 3 from [6], i.e. identify the dual of the space of Lipschitz psumming operators from a finite metric space to a Banach space, we will need to "reverse" the duality given by Theorem 4.3.…”

Section: Dualitymentioning

confidence: 99%

Duality for Lipschitz p-summing operators

Chávez-Domínguez

2011

Journal of Functional Analysis

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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.

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“…For the definition of Lipschitz p-summing mappings and the corresponding Pietsch Domination Theorem we refer to [10,4].…”

Section: Open Problemsmentioning

confidence: 99%

“…The measure µ above is also called a Pietsch measure for v. The Pietsch Domination Theorem has analogs in different contexts, including versions for classes of absolutely summing nonlinear operators (see, for example, [1,6,7,9,10,15,22]). Recently, in [4,21,23] the concept of abstract R-S-abstract p-summing mapping was introduced in such a way that several previous known versions of the Pietsch Domination Theorem can be regarded as particular instances of one single result.…”

Section: Introductionmentioning

confidence: 99%

When is the Haar measure a Pietsch measure for nonlinear mappings?

et al. 2012

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We show that, as in the linear case, the normalized Haar measure on a compact topological group G is a Pietsch measure for nonlinear summing mappings on closed translation invariant subspaces of C(G). This answers a question posed to the authors by J. Diestel. We also show that our result applies to several well-studied classes of nonlinear summing mappings. In the final section some problems are proposed.

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Lipschitz $p$-summing operators

Abstract: Abstract. The notion of Lipschitz p-summing operator is introduced. A nonlinear Pietsch factorization theorem is proved for such operators, and it is shown that a Lipschitz p-summing operator that is linear is a p-summing operator in the usual sense.

Cited by 71 publications

References 9 publications

Lipschitz compact operators

Lipschitz compact operators

Duality for Lipschitz p-summing operators

When is the Haar measure a Pietsch measure for nonlinear mappings?

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