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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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.

We introduce the concepts of Pełczyński's property ( ) of order and Pełczyński's property ( * ) of order . It is proved that, for each 1 < < ∞, the James -spaces enjoys Pełczyński's property ( * ) of order and the James * -spaces * (where * denotes the conjugate number of ) enjoys Pełczyński's property ( ) of order . We prove that both 1 ( ) ( a finite positive measure) and 1 enjoy a quantitative version of Pełczyński's property ( * ). K E Y W O R D SPełczyński's property ( ) of order , Pełczyński's property ( * ) of order , quantitative Pełczyński's property ( * ) of order M S C ( 2 0 1 0 ) 46-B

There are well-known relationships between compressed sensing and the geometry of the finite-dimensional p spaces. A result of Kashin and Temlyakov [20] can be described as a characterization of the stability of the recovery of sparse vectors via 1 -minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional 1 and 2 spaces, whereas a more recent result of Foucart, Pajor, Rauhut and Ullrich [16] proves an analogous relationship even for p spaces with p < 1. In this paper we prove what we call matrix or noncommutative versions of these results: we characterize the stability of low-rank matrix recovery via Schatten p-(quasi-)norm minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional Schatten p-spaces.