Displaying similar documents to “Absolutely continuous linear operators on Köthe-Bochner spaces”

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...

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...

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.

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...

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...

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,...

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then embeds in L(X,Y), and ℓ¹ embeds complementably in X γ Y * . Applications to embeddings of c₀ in various spaces of operators are given.

Stability of infinite ranges and kernels

K.-H. Förster, V. Müller (2006)

Studia Mathematica

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Let A(·) be a regular function defined on a connected metric space G whose values are mutually commuting essentially Kato operators in a Banach space. Then the spaces R ( A ( z ) ) and N ( A ( z ) ) ¯ do not depend on z ∈ G. This generalizes results of B. Aupetit and J. Zemánek.

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].

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 .

Besov spaces and 2-summing operators

M. A. Fugarolas (2004)

Colloquium Mathematicae

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Let Π₂ be the operator ideal of all absolutely 2-summing operators and let I m be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of I m . Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

Dunford-Pettis operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let L Φ ( X ) be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to L Φ ( X ) . In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to L ( X ) is ( τ ( L ( X ) , L ¹ ( X * ) ) , | | · | | Y ) -compact.

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 ∈...

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 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.

A new characterization of Anderson’s inequality in C 1 -classes

S. Mecheri (2007)

Czechoslovak Mathematical Journal

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Let be a separable infinite dimensional complex Hilbert space, and let ( ) denote the algebra of all bounded linear operators on into itself. Let A = ( A 1 , A 2 , , A n ) , B = ( B 1 , B 2 , , B n ) be n -tuples of operators in ( ) ; we define the elementary operators Δ A , B ( ) ( ) by Δ A , B ( X ) = i = 1 n A i X B i - X . In this paper, we characterize the class of pairs of operators A , B ( ) satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators A , B ( ) such that i = 1 n B i T A i = T implies i = 1 n A i * T B i * = T for all T 𝒞 1 ( ) (trace class operators). The main result is the equivalence between this property and the...

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...

Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the H calculus, H ( T ) , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes S p .

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 < ∞.

Non-hyperreflexive reflexive spaces of operators

Roman V. Bessonov, Janko Bračič, Michal Zajac (2011)

Studia Mathematica

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We study operators whose commutant is reflexive but not hyperreflexive. We construct a C₀ contraction and a Jordan block operator S B associated with a Blaschke product B which have the above mentioned property. A sufficient condition for hyperreflexivity of S B is given. Some other results related to hyperreflexivity of spaces of operators that could be interesting in themselves are proved.

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.

Bilinear operators associated with Schrödinger operators

Chin-Cheng Lin, Ying-Chieh Lin, Heping Liu, Yu Liu (2011)

Studia Mathematica

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Let L = -Δ + V be a Schrödinger operator in d and H ¹ L ( d ) be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by T ± ( f , g ) ( x ) = ( T f ) ( x ) ( T g ) ( x ) ± ( T f ) ( x ) ( T g ) ( x ) , where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from L p ( d ) × L q ( d ) to H ¹ L ( d ) for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails. ...

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.

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.