### Browsing by Author "Villanueva, Ignacio"

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Publication A composition theorem for multiple summing operators(Springer Wien, 2005-11) Villanueva, Ignacio; Pérez García, DavidWe prove that the composition S(u(1),..., u(n)) of a multilinear multiple 2-summing operator S with 2-summing linear operators u(j) is nuclear, generalizing a linear result of Grothendieck.Publication A decomposition theorem for polymeasures(Elsevier, 2007) Villanueva, Ignacio; Bombal Gordón, Fernando; Pérez García, DavidWe prove that every countably additive polymeasure can be decomposed in a unique way as the sum of a "discrete" polymeasure plus a "continuous" polymeasure. This generalizes a previous result of Saeki.Publication Characterization of dual mixed volumes via polymeasures(Elsevier, 2015) Jiménez, Carlos; Villanueva, IgnacioWe prove a characterization of the dual mixed volume in terms of functional properties of the polynomial associated with it. To do this, we use tools from the theory of multilinear operators on spaces of continuous functions. Along the way we reprove, with these same techniques, a recently found characterization of the dual mixed volume.Publication Completely continuous multilinear operators on C(K) spaces(America Mathematical Society, 2000) Villanueva, IgnacioGiven a k-linear operator T from a product of C(K) spaces into a Banach space X, our main result proves the equivalence between T being completely continuous, T having an X-valued separately omega* - omega* continuous extension to the product of the biduals and T having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to T being weakly compact, and that, for k > 1, T being weakly compact implies the conditions above but the converse fails.Publication Continuous valuations on the space of Lipschitz on the sphere(Elsevier, 2020-11-30) Colesanti, Andrea; Pagnini, Daniele; Tradacete Pérez, Pedro; Villanueva, IgnacioWe study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere Sn−1. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere. ContentsPublication Derivative and factorization of holomorphic functions(Elsevier, 2008-12-01) Bombal Gordón, Fernando; Gutiérrez, Joaquín M.; Villanueva, IgnacioLet E be a complex Banach space and denote by H b (E) the space of all holomorphic functions f:E→C of bounded type, that is, bounded on bounded sets. It is known that f∈H b (E) admits a factorization of the form f=g∘S , with S a compact linear operator and g a holomorphic function of bounded type if and only if the derivative df:E→E ∗ takes bounded sets of E into relatively compact sets of E ∗ . In the weakly compact case, a similar result was obtained by R. M. Aron and P. Galindo [Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 1, 181–192;]. In the paper under review, the authors extend these results to closed injective operator ideals and the associated families of bounded sets. Moreover, this main result is applied to give examples of factorization through operators belonging to important closed injective operator ideals.Publication Euclidean Distance Between Haar Orthogonal and Gaussian Matrices(Springer New York LLC, 2016) González Guillén, Carlos Eduardo; Palazuelos Cabezón, Carlos; Villanueva, IgnacioIn this work, we study a version of the general question of how well a Haar-distributed orthogonal matrix can be approximated by a random Gaussian matrix. Here, we consider a Gaussian random matrix (Formula presented.) of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix (Formula presented.). If (Formula presented.) denotes the vector formed by the first m-coordinates of the ith row of (Formula presented.) and (Formula presented.), our main result shows that the Euclidean norm of (Formula presented.) converges exponentially fast to (Formula presented.), up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm (Formula presented.) and we find a coupling that improves by a factor (Formula presented.) the recently proved best known upper bound on (Formula presented.). Our main result also has applications in Quantum Information Theory.Publication Extension of multilinear operators on Banach spaces(Universidad de Extremadura, Departamento de Matemáticas, 2001) Villanueva, Ignacio; Cabello Sánchez, Félix; Garcia, R.This paper considers the problem of extending multilinear forms on a Banach space X to a larger space Y containing it as a closed subspace. For instance, if X is a subspace of Y and X0 ! Y 0 extends linear forms, then the induced Nicodemi operators extend multilinear forms. It is shown that an extension operator X0 ! Y 0 exists if and only if X is locally complemented in Y . Also, these extension operators preserve the symmetry if and only if X is regular. Finally, multlinear characterizations are obtained of some classical Banach space properties (Dunford-Pettis, etc.) related to weak compactness in terms of operators having Z-valued Aron-Berner extensions.Publication Extensions of multilinear operators and Banach space properties(Cambridge University Press, 2003-06) Villanueva, Ignacio; Gutiérrez, Joaquín MA new characterization of the Dunford-Pettis property in terms of the extensions of multilinear operators to the biduals is obtained. For the first time, multilinear characterizations of the reciprocal Dunford-Pettis property and Pelczynski's property (V) are also found. Polynomial and holomorphic versions of these properties are given as well.Publication Factorizing multilinear operators on Banach spaces, C*-algebras and JB*-triples(Polish Acad Sciencies Inst Mathematics, 2009) Palazuelos Cabezón, Carlos; Peralta Pereira, Antonio Miguel; Villanueva, IgnacioIn recent papers, the Right mid the Strong* topologies have been. introduced and studied on general Banach spaces We characterize different types of continuity for multilinear operators (joint, uniform, etc) with respect to the above topologies We also study the relations between them. Finally, in the last section, we relate the joint Strong*-to-norm continuity of a multilinear operator T defined on C*-algebras (respectively, JB*-triples) to C*-summability (respectively, JB*-triple-summability).Publication Hahn–Banach extension of multilinear forms and summability(Academic Press, 2007-12-15) Jarchow , Hans; Palazuelos Cabezón, Carlos; Pérez García, David; Villanueva, IgnacioThe aim of this paper is to investigate close relations between the validity of Hahn–Banach extension theorems for multilinear forms on Banach spaces and summability properties of sequences from these spaces. A case of particular importance occurs when we consider Banach spaces which have the property that every bilinear form extends to any superspace.Publication Integral mappings between Banach spaces(Elsevier, 2003) Villanueva, IgnacioWe consider the classes of “Grothendieck-integral” (G-integral)and “Pietsch-integral” (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon-Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K,X) spaces and we also give a partial characterization of P-integral operators on C(K,X) spaces.Publication Integral multilinear forms on C(K,X) spaces(Springer Verlag, 2004) Villanueva, IgnacioWe use polymeasures to characterize when a multilinear form defined on a product of C(K, X) spaces is integral.Publication Integral operators on the product of C(K) spaces(Elsevier, 2001-12-01) Villanueva, Ignacio; Bombal Gordón, FernandoWe study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the representing polymeasure of the operator. Some applications are given. In particular, we characterize the Borel polymeasures that can be extended to a measure in the product ¾- algebra, generalizing previous results for bimeasures. We also give necessary conditions for the weak compactness of the extension of an integral multilinear operator on a product of C(K) spaces.Publication Joint system quantum descriptions arising from local quantumness(Springer, 2013) Cooney, Tom; Junge, Marius; Navascués Cobo, Miguel; Pérez García, David; Villanueva, IgnacioBipartite correlations generated by non-signalling physical systems that admit a finite-dimensional local quantum description cannot exceed the quantum limits, i.e., they can always be interpreted as distant measurements of a bipartite quantum state. Here we consider the effect of dropping the assumption of finite dimensionality. Remarkably, we find that the same result holds provided that we relax the tensor structure of space-like separated measurements to mere commutativity. We argue why an extension of this result to tensor representations seems unlikely.Publication Local structure and copies of c(0) and l(1) in the tensor product of Banach spaces(Sociedad Matemática Mexicana, 2004) Bombal Gordón, Fernando; Fernández Unzueta, M.; Villanueva, IgnacioWe use well known properties of the tensor product of l(p)-spaces to study the local structure of projective and injective tensor products of Banach spaces. In particular we give a simple proof of the fact that the injective (resp. projective) tensor product of infinite dimensional Banach spaces contains the l(infinity)(n)'s (resp., l(1)(n)'s) uniformly complemented. We then refine the previous arguments to give criteria for obtaining copies (complemented or not) of c(0) in the injective tensor product of Banach spaces, and complemented copies of l(1) in the projective tenser product of Banach spaces.Publication Measures on the product of compact spaces(Springer, 2002) Villanueva, IgnacioIf K is an uncountable metrizable compact space, we prove a ‘factorization’ result for a wide variety of vector valued Borel measures μ defined on Kn. This result essentially says that for every such measure μ there exists a measure μ0 defined on K such that the measure μ of a product A1 ×• • •×An of Borel sets of K equals the measure μ0 of the intersection A01 \• • •\A0n, where the A0j’s are certain transforms of the Ai’s.Publication Multilinear extensions of Grothendieck’s theorem(Oxford University Press, 2004) Villanueva, Ignacio; Bombal Gordón, Fernando; Pérez García, DavidWe introduce a new class of multilinear p-summing operators, which we call multiple p-summing. Using them, we can prove several multilinear generalizations of Grothendieck’s “fundamental theorem of the metric theory of tensor products”. Several applications and improvements of previous results are given.Publication Multilinear operators on C(K,X) spaces(Mathematical Institute of the Academy of Sciences of the Czech Republic, 2004) Villanueva, IgnacioGiven Banach spaces X, Y and a compact Hausdor_ space K, we use polymeasures to give necessary conditions for a multilinear operator from C(K;X) into Y to be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for X to have the Schur property (resp. to contain no copy of c0), and for K to be scattered. This extends results concerning linear operators.Publication Multilinear operators on spaces of continuous functions(Faculty of Mathematics and Computer Science of Adam Mickiewicz University, 1998) Villanueva, Ignacio; Bombal Gordón, FernandoLet E1, . . . ,Ed be Banach spaces such that all linear operators from Ei into E_j (i 6= j) are weakly compact. The authors show that every continuous d-linear operator T on E1 × • • • × Ed to a Banach space F possesses a unique bounded multilinear extension T__ : E__ 1 × • • • × E__ d ! F__ that is !_ − !_-separately continuous and kT__k = kTk. In particular, existence of unique continuous multilinear extensions from C(K1)×• • •× C(Kd) (Ki – Hausdorff compact spaces) to C(K1)__×• • •×C(Kd)__ that are separately weak_-continuous is established. As a corollary, integral representations with respect to polymeasures for multilinear mappings on C(K1)×• • •×C(Kd) into a Banach space are found. The results generalize a theorem due to Pelczynsky about multilinear extensions from C(K1) × • • • × C(Kd) to the Cartesian product of the spaces of bounded Baire functions on Ki.

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