Absolutely Summing Operator

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Dumitru Popa - One of the best experts on this subject based on the ideXlab platform.

Narcisse Randrianantoanina - One of the best experts on this subject based on the ideXlab platform.

  • Compact range property and Operators on C^*-algebras
    Proceedings of the American Mathematical Society, 2000
    Co-Authors: Narcisse Randrianantoanina
    Abstract:

    We prove that a Banach space E has the compact range property (CRP) if and only if, for any given C*-algebra A, every Absolutely Summing Operator from A into E is compact. Related results for p-Summing Operators (0 < p < 1) are also discussed as well as Operators on non-commutative Llspaces and C*-Summing Operators.

  • Compact range property and Operators on $C^*$-algebras
    arXiv: Functional Analysis, 2000
    Co-Authors: Narcisse Randrianantoanina
    Abstract:

    We prove that a Banach space $E$ has the compact range property (CRP) if and only if for any given $C^*$-algebra $\cal A$, every Absolutely Summing Operator from $\cal A$ into $E$ is compact.

  • Absolutely Summing Operators on non-commutative C ∗ -algebras and applications
    1999
    Co-Authors: Narcisse Randrianantoanina
    Abstract:

    Abstract. Let E be a Banach space that does not contain any copy of ℓ 1 and A be a non commutative C ∗-algebra. We prove that every Absolutely Summing Operator from A into E ∗ is compact, thus answering a question of Pe̷lczynski. As application, we show that if G is a compact metrizable abelian group and Λ is a Riesz subset of its dual then every countably additive A ∗-valued measure with bounded variation and whose Fourier transform is supported by Λ has relatively compact range. Extensions of the same result to symmetric spaces of measurable Operators are also presented. 1

  • Factorization of Operators on $C^*$-algebras
    arXiv: Functional Analysis, 1997
    Co-Authors: Narcisse Randrianantoanina
    Abstract:

    Let $A$ be a $C^*$-algebra. It is shown that every Absolutely Summing Operator from $A$ into $\ell_2$ factors through a Hilbert space Operator that belongs to the 4-Schatten- von Neumann class. We also provide finite dimensinal examples that show that one can not improve the 4-Schatten-von Neumann class to $p$-Schatten von Neumann class for any $p 0$, the $\epsilon$-capacity of the image of the unit ball of $A$ under $T$ does not exceed $N(\epsilon)$. This aswers positively a question raised by Pe\l czynski.

  • Absolutely Summing Operators on non commutative $C^*$-algebras and applications
    arXiv: Functional Analysis, 1995
    Co-Authors: Narcisse Randrianantoanina, Vern I. Paulsen
    Abstract:

    Let E be a Banach space that does not contain any copy of ' 1 andA be a non commutative C -algebra. We prove that every Absolutely Summing Operator fromA into E is compact, thus answering a question of Pe lczy nski. As application, we show that if G is a compact metrizable abelian group and is a Riesz subset of its dual then every countably additive A -valued measure with bounded variation and whose Fourier transform is supported by has relatively compact range. Extensions of the same result to sym- metric spaces of measurable Operators are also presented.

José Rodríguez - One of the best experts on this subject based on the ideXlab platform.

  • On the equivalence of McShane and Pettis integrability in non-separable Banach spaces
    Journal of Mathematical Analysis and Applications, 2008
    Co-Authors: José Rodríguez
    Abstract:

    Abstract We show that McShane and Pettis integrability coincide for functions f : [ 0 , 1 ] → L 1 ( μ ) , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelof determined Banach space X, a scalarly null (hence Pettis integrable) function h : [ 0 , 1 ] → X and an Absolutely Summing Operator u from X to another Banach space Y such that the composition u ○ h : [ 0 , 1 ] → Y is not Bochner integrable; in particular, h is not McShane integrable.

  • Absolutely Summing Operators and integration of vector-valued functions
    Journal of Mathematical Analysis and Applications, 2006
    Co-Authors: José Rodríguez
    Abstract:

    AbstractLet (Ω,Σ,μ) be a complete probability space and u:X→Y an Absolutely Summing Operator between Banach spaces. We prove that for each Dunford integrable (i.e., scalarly integrable) function f:Ω→X the composition u○f is scalarly equivalent to a Bochner integrable function. Such a composition is shown to be Bochner integrable in several cases, for instance, when f is properly measurable, Birkhoff integrable or McShane integrable, as well as when X is a subspace of an Asplund generated space or a subspace of a weakly Lindelöf space of the form C(K). We also study the continuity of the composition Operator f↦u○f. Some other applications are given

Randrianantoanina Narcisse - One of the best experts on this subject based on the ideXlab platform.

Popa Dumitru - One of the best experts on this subject based on the ideXlab platform.