### Browsing by Author "Bombal Gordón, Fernando"

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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 A double-dual characterization of Rosenthal and semitauberian operators(Royal Irish Academy, 1995-08) Bombal Gordón, Fernando; Hernando Boto, BeatrizTauberian operators between Banach spaces were introduced by N. Kalton and A. Wilansky [Proc. Amer. Math. Soc. 57 (1976), no. 2, 251–255;] as those operators T:X→Y whose second conjugate satisfies (T **) −1 Y⊂X . Note that ker(T **)⊂X is a necessary, but not sufficient, condition for T to be Tauberian. These operators were characterized by the "preservation'' of weakly convergent sequences: a bounded sequence (x n ) in X admits a weakly convergent subsequence whenever (Tx n ) is weakly convergent. This is a key fact in the study of the factorization due to W. J. Davis et al. [J. Functional Analysis 17 (1974), 311–327;], which allows one to factorize every weakly compact operator through a reflexive space. Semi-Tauberian operators are defined as those operators that preserve weakly Cauchy sequences, and Rosenthal operators are defined by the opposite property: they take bounded sequences into sequences admitting weakly Cauchy subsequences. The class of semi-Tauberian operators is formally similar to that of Tauberian operators. For example, it was shown by the reviewer and V. M. Onieva [Proc. Amer. Math. Soc. 108 (1990), no. 2, 399–405;] that T is Tauberian [semi-Tauberian] if and only if for every compact operator K the kernel ker(T+K) is reflexive [contains no copy of l 1 ]. In the present paper several characterizations of semi-Tauberian operators and Rosenthal operators are presented. The most remarkable result emphasizes the differences with Tauberian operators: In the case X separable, T:X→Y is semi-Tauberian if and only if ker(T **) is contained in the class B 1 (X) of vectors in X ∗∗ which are limits of weakly Cauchy sequences in X ; and this is equivalent to either (T **) −1 B 1 (Y)⊂B 1 (X) , or (T **) −1 B 1 (Y)⊂X . Analogous results are proved in the case X non-separable, in terms of suitable classes of vectors in X **.Publication A new operator characterization of the Dunford-Pettis property(Universidad de Extremadura, Departamento de Matemáticas, 1987) Bombal Gordón, FernandoPublication A normability criterion.(Spanish: Un criterio de normabilidad)(Consejo Superior de Investigaciones Científicas, 1969) Bombal Gordón, FernandoThe author proves that a metrizable locally convex space E is a normable space if and only if there exists a scalar 1 such that the linear mapping x→1x from E to E is contractive for every metric compatible with the topology of E .Publication Alexander Grothendieck's work on functional analysis.(World Scientific, 2007) Bombal Gordón, Fernando; Velasco, M. V.; Rodríguez-Palacios, A.After a short biographical sketch, the author describes some aspects of Alexander Grothendieck's contributions to functional analysis. He focuses on the following topics: tensor products, the approximation property, nuclear and integral operators, nuclear spaces, the Dunford-Pettis property, and Grothendieck's ``Résumé'' paper. An appendix lists all the functional analytic papers by Grothendieck.Publication Algunos espacios de Banach de funciones vectoriales.(Universidade do Minho, 1987) Bombal Gordón, FernandoPublication Almost convergent vector-valued functions. (Spanish: Funciones vectoriales casi-convergentes).(Springer, 1975) Bombal Gordón, Fernando; Vera Boti, GabrielIn this paper the authors extend the concept of almost convergence to vector-valued functions on a semigroup having a left invariant mean. They study the space of almost convergent functions and obtain characterization theorems similar to known theorems about real functions. They also establish the existence of a unique invariant mean on vector-valued weakly almost periodic functions.Publication An extension of sequential spaces. (Spanish: Una extensión de los espacios secuenciales).(Real Academia de Ciencias Exactas, Físicas y Naturales, 1979) Bombal Gordón, Fernando; Rodríguez Marín, LuisOn considère un espace topologique muni aussi d'une bornolgie telle que tout point adhérent à un ensemble A⊂E soit limite d'une suite généralisée bornée de points de A. Cette structure est caractérisée par le fait que toute application de E dans un espace topologique, continue en x sur les ensembles bornés, est continue en x. A la bornologie correspond une seconde topologie, plus fine, telle que les applications continues par rapport à cette topologie soient exactement celles qui sont continues dans les ensembles bornés par rapport à la topologie initiale. Si E est aussi un espace vectoriel, avec des relations de compatibilité, si la bornologie contient les suites et tout ensemble bornivore est un voisinage de 0, alors la topologie de E est de type séquentiel.Publication Baltasar Rodríguez-Salinas, in memoriam(Real Sociedad Matemática Española, 2007) Bombal Gordón, FernandoPublication Bourbaki(Real Academia de Ciencias Exactas, Físicas y Naturales., 1998) Bombal Gordón, FernandoPublication Characterization of some classes of operators on spaces of vector-valued continuous functions(Cambridge University Press, 1985) Bombal Gordón, Fernando; Cembranos, PilarLet $K$K be a compact Hausdorff space and $E$E, $F$F Banach spaces with $L(E,F)$L(E,F) the space of bounded linear operators from $E$E into $F$F. If $C(K,E)$C(K,E) is the space of all continuous functions from $K$K into $E$E equipped with the sup-norm, then every operator $T\in L(C(K,E),F)$T∈L(C(K,E),F) has a representing measure $m$m of bounded semivariation on the Borel sets of $K$K with values in $L(E,F'')$L(E,F′′) such that $TF=\int_Kf\,dm$TF=∫Kfdm. If $T$T is a weakly compact operator, then $m$m has values in $L(E,F)$L(E,F), $m(E)$m(E) is weakly compact for each Borel set $E$E, and the semivariation of $m$m is continuous at $\varphi$φ. It is known that the converse of this statement does not hold in general, but does hold under additional assumptions. In particular, the authors show that the converse holds if $K$K is a dispersed space. They also show that, in a certain sense, the assumption that $K$K is a dispersed space is necessary; that is, if the converse of the statement above holds for every pair of Banach spaces $E,F$E,F then $K$K must be a dispersed space. A similar result holds for the class of unconditionally converging, Dunford-Pettis or Dieudonne operators.Publication Continuous multilinear operators on C(K) spaces and polymeasures(Universidad de Extremadura, Departamento de Matemáticas, 2007) Bombal Gordón, FernandoThis paper is a survey of the results of Bombal and I. Villanueva Díez [Funct. Approx. Comment. Math. 26 (1998), 117–126; Bombal, D. Pérez-García and Villanueva Díez [Q. J. Math. 55 (2004), no. 4, 441–450; and Pérez-García and Villanueva Díez [Ark. Mat. 42 (2004), no. 1, 153–171; The results surveyed concern Riesz-type integral representations of continuous multilinear operators on C(K) -spaces by polymeasures.Publication Correction to: "Means in locally convex spaces and semi-reflexivity''(Spanish: Correcciones a:"Medias en espacios localmente convexos y semi-reflexividad")(Springer, 1975) Bombal Gordón, Fernando; Vera Boti, GabrielThe authors indicate some mistakes which appear in the article cited in the title. They correct Corollary 29, Proposition 30 and Theorem 31 with the additional condition which is quite exigent and difficult to apply: the restriction of the elements of E ′ to E B separates points of E BPublication 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 Dieudonné operators on C(K,E)(Polish Academy of Sciences, 1986) Bombal Gordón, Fernando; Cembranos, PilarA Banach space operator is called a Dieudonné operator if it maps weakly Cauchy sequences to weakly convergent sequences. A space E is said to have property (D) if, whenever K is a compact Hausdorff space and T is an operator from C(K,E) into a space F , T is a Dieudonné operator if and only if its representing measure is both strongly additive and has for its values Dieudonné operators from E into F . The purpose of this paper is to show that if E ∗ has the Radon-Nikodým property then E has (D) if and only if E ∗∗ has the Radon-Nikodým property.Publication Differentiable function spaces - Preliminary-Report(American Mathematical Society, 1976) Llavona, José G.; Bombal Gordón, FernandoPublication Distinguished subsets and complemented copies of co in vector sequence spaces(Universidad de Extremadura, Departamento de Matemáticas, 1990) Bombal Gordón, FernandoPublication Distinguished subsets in vector sequence spaces(Elsevier, 1992) Bombal Gordón, FernandoWe characterize several classes of subsets in the Λ-sum (ΣEn)Λ of a sequence (En) of Banach spaces (where A is a Banach sequence ideal space), in terms of the corresponding classes in the spaces En. Several stability results are obtained.Publication Dos nuevas clases de espacios de Banach(Universidad de La Laguna. Secretariado de Publicaciones, 1990) Bombal Gordón, FernandoPublication El Análisis Infinitesimal en Euler(Instituto de España, 2009) Bombal Gordón, Fernando; Galindo Tixaire, Alberto; López Pellicer, Manuel