Displaying similar documents to “On operators on L 0

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...

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...

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

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Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let ( X , | | · | | X ) and ( Y , | | · | | Y ) be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if | | T ( 1 A f ) | | Y 0 whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

Young's (in)equality for compact operators

Gabriel Larotonda (2016)

Studia Mathematica

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If a,b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then λ k ( | a b * | ) λ k ( 1 / p | a | p + 1 / q | b | q ) for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young’s inequality if and only if | a | p = | b | q .

Multiple summing operators on l p spaces

Dumitru Popa (2014)

Studia Mathematica

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We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of l p spaces. This characterization is used to show that multiple s-summing operators on a product of l p spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators T : l 4 / 3 × l 4 / 3 l such that none of the associated linear operators...

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.

Weakly precompact operators on C b ( X , E ) with the strict topology

Juliusz Stochmal (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C b ( X , E ) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators T : C b ( X , E ) F . In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator T : C b ( X , E ) F is weakly precompact.

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].

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,...

Commutators on ( q ) p

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on X = ( q ) p with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.

Regularity of domains of parameterized families of closed linear operators

Teresa Winiarska, Tadeusz Winiarski (2003)

Annales Polonici Mathematici

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The purpose of this paper is to provide a method of reduction of some problems concerning families A t = ( A ( t ) ) t of linear operators with domains ( t ) t to a problem in which all the operators have the same domain . To do it we propose to construct a family ( Ψ t ) t of automorphisms of a given Banach space X having two properties: (i) the mapping t Ψ t is sufficiently regular and (ii) Ψ t ( ) = t for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition...

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

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Given two n-tuples a = ( a 1 , . . . , a n ) and b = ( b 1 , . . . , b n ) of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that E a j = b j for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in A n .

Corrigendum to “Commutators on ( q ) p ” (Studia Math. 206 (2011), 175-190)

Dongyang Chen, William B. Johnson, Bentuo Zheng (2014)

Studia Mathematica

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We give a corrected proof of Theorem 2.10 in our paper “Commutators on ( q ) p ” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.

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.

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 < ∞.

Application of ( L ) sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&amp;#039;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...

On the (C,α) Cesàro bounded operators

Elmouloudi Ed-dari (2004)

Studia Mathematica

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For a given linear operator T in a complex Banach space X and α ∈ ℂ with ℜ (α) > 0, we define the nth Cesàro mean of order α of the powers of T by M α = ( A α ) - 1 k = 0 n A n - k α - 1 T k . For α = 1, we find M ¹ = ( n + 1 ) - 1 k = 0 n T k , the usual Cesàro mean. We give necessary and sufficient conditions for a (C,α) bounded operator to be (C,α) strongly (weakly) ergodic.

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

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We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to Σ α , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a Π α uniformization which is the graph of a Σ α -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with G δ sections. ...

Effective decomposition of σ-continuous Borel functions

Gabriel Debs (2014)

Fundamenta Mathematicae

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We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ( A ) n ω of X such that f | A is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

On a binary relation between normal operators

Takateru Okayasu, Jan Stochel, Yasunori Ueda (2011)

Studia Mathematica

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The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that E A ( Δ ) T * E B ( Δ ) T for all Borel subset Δ of the complex plane ℂ, where E A and E B are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is...

Order bounded composition operators on the Hardy spaces and the Nevanlinna class

Nizar Jaoua (1999)

Studia Mathematica

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We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces H p 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with X = H p or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into L h . We give...

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.

The ideal of p-compact operators: a tensor product approach

Daniel Galicer, Silvia Lassalle, Pablo Turco (2012)

Studia Mathematica

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We study the space of p-compact operators, p , using the theory of tensor norms and operator ideals. We prove that p is associated to / d p , the left injective associate of the Chevet-Saphar tensor norm d p (which is equal to g p ' ' ). This allows us to relate the theory of p-summing operators to that of p-compact operators. Using the results known for the former class and appropriate hypotheses on E and F we prove that p ( E ; F ) is equal to q ( E ; F ) for a wide range of values of p and q, and show that our results...

A singular initial value problem for the equation u ( n ) ( x ) = g ( u ( x ) )

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

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We consider the problem of the existence of positive solutions u to the problem u ( n ) ( x ) = g ( u ( x ) ) , u ( 0 ) = u ' ( 0 ) = . . . = u ( n - 1 ) ( 0 ) = 0 (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition δ 1 / s [ s / g ( s ) ] 1 / n d s < is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let i : L ( X ) L ¹ ( X ) stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then T i : L ( X ) Y is a weakly compact operator. Moreover, we obtain that...

- and -hypoellipticity of partial differential operators with constant Colombeau coefficients

Claudia Garetto (2010)

Banach Center Publications

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We provide a deep investigation of the notions of - and -hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and -hypoellipticity are given

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 ∈...

Some properties and applications of equicompact sets of operators

E. Serrano, C. Piñeiro, J. M. Delgado (2007)

Studia Mathematica

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Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence ( x k ( n ) ) such that ( T x k ( n ) ) is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness...

Extension operators on balls and on spaces of finite sets

Antonio Avilés, Witold Marciszewski (2015)

Studia Mathematica

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We study extension operators between spaces of continuous functions on the spaces σ ( 2 X ) of subsets of X of cardinality at most n. As an application, we show that if B H is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator T : C ( λ B H ) C ( μ B H ) .