Displaying similar documents to “Opérateurs sur un espace L 1

Compact operators whose adjoints factor through subspaces of l p

Deba P. Sinha, Anil K. Karn (2002)

Studia Mathematica

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For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if K n = 1 α x : α B a l l ( l p ' ) , where p’ = p/(p-1) and x l p s ( X ) . An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering x l p w ( X ) . It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of l p in a particular manner. The normed operator ideals ( K p , κ p ) of p-compact operators and ( W p , ω p ) of weakly p-compact operators, arising from these factorizations,...

Application of ( L ) sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H'michane, Aziz Elbour (2016)

Mathematica Bohemica

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The paper contains some applications of the notion of Ł sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order ( L ) -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an ( L ) sets. As a sequence characterization of such operators, we see that an operator T : X E from a Banach space into a Banach lattice is order Ł -Dunford-Pettis, if and only if | T ( x n ) | 0 for σ ( E , E ' ) for every...

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let L w * ( E * , F ) (resp. K w * ( E * , F ) ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of K w * ( E * , F ) such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then K w * ( E * , F ) has property (wV*). Suppose...

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space K w * ( X * , Y ) , as well as the complementation of the space K w * ( X * , Y ) of w*-w compact operators in the space L w * ( X * , Y ) of w*-w operators from X* to Y.

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak (2005)

Banach Center Publications

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Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space ( X , | | · | | X ) let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function | | f ( · ) | | X belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let D u ( = f E ( X ) : | | f ( · ) | | X u ) stand for the order interval in E(X). For a real Banach space ( Y , | | · | | Y ) a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

Factorization of CP-rank- 3 completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

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A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = B B . If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A . In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3 . Failure of this algorithm implies that A does not have cp-rank 3 . Our motivation stems from the question if there exist three nonnegative polynomials of degree at...

On the real X -ranks of points of n ( ) with respect to a real variety X n

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let  X n be an integral and non-degenerate m -dimensional variety defined over . For any P n ( ) the real X -rank r X , ( P ) is the minimal cardinality of S X ( ) such that P S . Here we extend to the real case an upper bound for the X -rank due to Landsberg and Teitler.

Spaces of compact operators on C ( 2 × [ 0 , α ] ) spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

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We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces 2 × [ 0 , α ] , the topological products of Cantor cubes 2 and intervals of ordinal numbers [0,α].

Simultaneous stabilization in A ( )

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra A ( ) . Invertible pairs ( f j , g j ) , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that α f j + β g j is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since A ( ) has stable rank two, we are faced here with a different...

The 4-string braid group B 4 has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

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We prove that the braid group B 4 on 4 strings, its central quotient B 4 / z , and the automorphism group Aut ( F 2 ) of the free group F 2 on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group B 4 is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

Property ( 𝐰𝐋 ) and the reciprocal Dunford-Pettis property in projective tensor products

Ioana Ghenciu (2015)

Commentationes Mathematicae Universitatis Carolinae

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A Banach space X has the reciprocal Dunford-Pettis property ( R D P P ) if every completely continuous operator T from X to any Banach space Y is weakly compact. A Banach space X has the R D P P (resp. property ( w L ) ) if every L -subset of X * is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product X π Y has property ( w L ) when X has the R D P P , Y has property ( w L ) , and L ( X , Y * ) = K ( X , Y * ) .

On strongly l p -summing m-linear operators

Lahcène Mezrag (2008)

Colloquium Mathematicae

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We introduce and study a new concept of strongly l p -summing m-linear operators in the category of operator spaces. We give some characterizations of this notion such as the Pietsch domination theorem and we show that an m-linear operator is strongly l p -summing if and only if its adjoint is l p -summing.

Cardinalities of DCCC normal spaces with a rank 2-diagonal

Wei-Feng Xuan, Wei-Xue Shi (2016)

Mathematica Bohemica

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A topological space X has a rank 2-diagonal if there exists a diagonal sequence on X of rank 2 , that is, there is a countable family { 𝒰 n : n ω } of open covers of X such that for each x X , { x } = { St 2 ( x , 𝒰 n ) : n ω } . We say that a space X satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of X is countable. We mainly prove that if X is a DCCC normal space with a rank 2-diagonal, then the cardinality of X is at most 𝔠 . Moreover, we prove that if X is a first...

On coincidence of Pettis and McShane integrability

Marián J. Fabián (2015)

Czechoslovak Mathematical Journal

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R. Deville and J. Rodríguez proved that, for every Hilbert generated space X , every Pettis integrable function f : [ 0 , 1 ] X is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [ 0 , 1 ] into X , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [ 0 , 1 ] (mostly) into C ( K ) spaces. We focus in more detail on...

The natural operators T | f T * T r * and T | f Λ ² T * T r *

W. M. Mikulski (2002)

Colloquium Mathematicae

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Let r and n be natural numbers. For n ≥ 2 all natural operators T | f T * T r * transforming vector fields on n-manifolds M to 1-forms on T r * M = J r ( M , ) are classified. For n ≥ 3 all natural operators T | f Λ ² T * T r * transforming vector fields on n-manifolds M to 2-forms on T r * M are completely described.

Representing non-weakly compact operators

Manuel González, Eero Saksman, Hans-Olav Tylli (1995)

Studia Mathematica

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For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of L 1 and C(0,1), but R(L(E)/W(E)) identifies isometrically...

2-summing multiplication operators

Dumitru Popa (2013)

Studia Mathematica

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Let 1 ≤ p < ∞, = ( X ) n be a sequence of Banach spaces and l p ( ) the coresponding vector valued sequence space. Let = ( X ) n , = ( Y ) n be two sequences of Banach spaces, = ( V ) n , Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator M : l p ( ) l q ( ) by M ( ( x ) n ) : = ( V ( x ) ) n . We give necessary and sufficient conditions for M to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

Multiplication operators on L ( L p ) and p -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on L ( L p ) , 1<p< , i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f p - s t r i c t l y s i n g u l a r o p e r a t o r s , a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l p - s t r i c t l y s i n g u l a r o p e r a t o r s o n Lp . T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r T o n Lp , 1<p<2 , i s p - s t r i c t l y s i n g u l a r a n d T|X i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e X o f Lp , t h e n X e m b e d s i n t o Lr f o r a l l r<2 , b u t X n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e . It is also shown that if T is convolution by a biased coin on L p of the Cantor group, 1 p < 2 , and T | X is an isomorphism for some reflexive subspace X of L p , then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.

Generalized Cesàro operators on certain function spaces

Sunanda Naik (2010)

Annales Polonici Mathematici

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Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators b , c is bounded on the Dirichlet spaces p , a . We also give a short and direct proof of boundedness of b , c on the Hardy space H p for 1 < p < ∞.

The natural linear operators T * T T ( r )

J. Kurek, W. M. Mikulski (2003)

Colloquium Mathematicae

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For natural numbers n ≥ 3 and r a complete description of all natural bilinear operators T * × f T ( 0 , 0 ) T ( 0 , 0 ) T ( r ) is presented. Next for natural numbers r and n ≥ 3 a full classification of all natural linear operators T * | f T T ( r ) is obtained.

Some duality results on bounded approximation properties of pairs

Eve Oja, Silja Treialt (2013)

Studia Mathematica

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The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair ( X * , Y ) has the λ-bounded approximation property. Then there exists a net ( S α ) of finite-rank operators on X such that S α ( Y ) Y and | | S α | | λ for all α, and ( S α ) and ( S * α ) converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.