Displaying similar documents to “Asymptotic behaviour of the iterates of nonnegative operators on a Banach lattice”

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

Similarity:

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.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

Similarity:

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.

The extension of the Krein-Šmulian theorem for order-continuous Banach lattices

Antonio S. Granero, Marcos Sánchez (2008)

Banach Center Publications

Similarity:

If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) d ̂ ( c o ¯ w * ( K ) , X ) 2 d ̂ ( K , X ) and, if K ∩ X is w*-dense in K, then d ̂ ( c o ¯ w * ( K ) , X ) = d ̂ ( K , X ) ; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then d ̂ ( c o ¯ w * ( K ) , X ) = d ̂ ( K , X ) ; (iii) if X has a 1-symmetric basis, then d ̂ ( c o ¯ w * ( K ) , X ) = d ̂ ( K , X ) .

Some properties and applications of equicompact sets of operators

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

Studia Mathematica

Similarity:

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...

Multiple summing operators on l p spaces

Dumitru Popa (2014)

Studia Mathematica

Similarity:

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

Similarity:

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...

Representing non-weakly compact operators

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

Studia Mathematica

Similarity:

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...

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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.

-vectors and boundedness

Jan Stochel, F. H. Szafraniec (1997)

Annales Polonici Mathematici

Similarity:

The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its -vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

Envelope functions and asymptotic structures in Banach spaces

Bünyamin Sarı (2004)

Studia Mathematica

Similarity:

We introduce a notion of disjoint envelope functions to study asymptotic structures of Banach spaces. The main result gives a new characterization of asymptotic- p spaces in terms of the p -behavior of “disjoint-permissible” vectors of constant coefficients. Applying this result to Tirilman spaces we obtain a negative solution to a conjecture of Casazza and Shura. Further investigation of the disjoint envelopes leads to a finite-representability result in the spirit of the Maurey-Pisier...

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

Elói Medina Galego (2011)

Colloquium Mathematicae

Similarity:

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].

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

Similarity:

This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form F ( x ) : = F ̃ x ( 2 n + 1 ) where F ̃ : X 2 n + 1 Y is a continuous (2n+1)-linear operator.

Note on distortion and Bourgain ℓ₁-index

Anna Maria Pelczar (2009)

Studia Mathematica

Similarity:

Relations between different notions measuring proximity to ℓ₁ and distortability of a Banach space are studied. The main result states that a Banach space all of whose subspaces have Bourgain ℓ₁-index greater than ω α , α < ω₁, contains either an arbitrarily distortable subspace or an α -asymptotic subspace.

Compact operators whose adjoints factor through subspaces of l p

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

Studia Mathematica

Similarity:

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,...

Stability of infinite ranges and kernels

K.-H. Förster, V. Müller (2006)

Studia Mathematica

Similarity:

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.

Application of ( L ) sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&amp;#039;michane, Aziz Elbour (2016)

Mathematica Bohemica

Similarity:

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...

Operator Lipschitz functions on Banach spaces

Jan Rozendaal, Fedor Sukochev, Anna Tomskova (2016)

Studia Mathematica

Similarity:

Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form | | f ( B ) S - S f ( A ) | | ( X , Y ) c o n s t | | B S - S A | | ( X , Y ) for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on X = p and Y = q for p < q. We also study the estimate above in the setting of Banach...

Semilinear perturbations of Hille-Yosida operators

Horst R. Thieme, Hauke Vosseler (2003)

Banach Center Publications

Similarity:

The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), u ( 0 ) = x D ( A ) ¯ , with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, B ( λ - A ) - 1 x = λ 0 e - λ t V ( s ) x d s , where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on [ 0 , ) × D ( A ) ¯ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤...

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

Marian Nowak (2005)

Banach Center Publications

Similarity:

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 ∈...

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

Similarity:

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 .

On Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2006)

Studia Mathematica

Similarity:

We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type but not conversely. Moreover, -spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will...

On the (C,α) Cesàro bounded operators

Elmouloudi Ed-dari (2004)

Studia Mathematica

Similarity:

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.

Essentially Incomparable Banach Spaces of Continuous Functions

Rogério Augusto dos Santos Fajardo (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We construct, under Axiom ♢, a family ( C ( K ξ ) ) ξ < 2 ( 2 ω ) of indecomposable Banach spaces with few operators such that every operator from C ( K ξ ) into C ( K η ) is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size 2 ω instead of 2 ( 2 ω ) .

The natural linear operators T * T T ( r )

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

Colloquium Mathematicae

Similarity:

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.

On the compact approximation property

Vegard Lima, Åsvald Lima, Olav Nygaard (2004)

Studia Mathematica

Similarity:

We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net ( S γ ) of compact operators on X such that s u p γ | | S γ T | | | | T | | and S γ I X in the strong operator topology. Similar results for dual spaces are also proved.

Subharmonic functions in sub-Riemannian settings

Andrea Bonfiglioli, Ermanno Lanconelli (2013)

Journal of the European Mathematical Society

Similarity:

In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution Γ . These characterizations are based on suitable average operators on the level sets of Γ . Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic...

Some duality results on bounded approximation properties of pairs

Eve Oja, Silja Treialt (2013)

Studia Mathematica

Similarity:

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.

Dunford-Pettis operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

Similarity:

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.

On the rate of convergence of the Bézier-type operators

Grażyna Anioł (2006)

Bollettino dell'Unione Matematica Italiana

Similarity:

For bounded functions f on an interval I , in particular, for functions of bounded p-th power variation on I there is estimated the rate of pointwise convergence of the Bezier-type modification of the discrete Feller operators. In the main theorem the Chanturiya modulus of variation is used.