Disjointly homogeneous Banach lattices: Duality and complementation (original) (raw)
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Disjointly homogeneous Banach lattices and applications
This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.
Disjointly homogeneous Banach lattices and compact products of operators
Journal of Mathematical Analysis and …, 2009
The notion of disjointly homogeneous Banach lattice is introduced. In these spaces every two disjoint sequences share equivalent subsequences. It is proved that on this class of Banach lattices the product of a regular AM-compact and a regular disjointly strictly singular operators is always a compact operator.
Disjointly non-singular operators on Banach lattices
Journal of Functional Analysis, 2021
An operator T from a Banach lattice E into a Banach space is disjointly non-singular (DN-S, for short) if no restriction of T to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for DN-S operators, including a perturbative characterization. For E = L p (1 < p < ∞) we improve the results, and we show that the DN-S operators have a different behavior in the cases p = 2 and p = 2. As an application we prove that the strongly embedded subspaces of L p form an open subset in the set of all closed subspaces.
Revista Matemática Complutense, 2020
We show that c 0 is not a projective Banach lattice, answering a question of B. de Pagter and A. Wickstead. .On the other hand, we show that c 0 is complemented in the free Banach lattice generated by itself (seen as a Banach space). As a consequence, the free Banach lattice generated by c 0 is not projective.
Reflexivity and the Separable Quotient Problem for a Class of Banach Spaces
Bulletin of the Polish Academy of Sciences Mathematics
Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or the norm of E is order continuous. Then X is reflexive iff X* contains no isomorphic copy of \ell_1 iff for every n ≥ 1, the nth dual X^(n) of X contains no isomorphic copy of \ell_1 iff X has no quotient isomorphic to c_0 and X does not have a subspace isomorphic to \ell_1 (Theorem 2). This is an extension of the results obtained earlier by Lozanovski˘ i, Tzafriri, Meyer-Nieberg, and Diaz and Fern´andez. The theorem is applied to show that many Banach spaces possess separable quotients isomorphic to one of the following spaces: c_0, \ell_1, or a reflexive space with a Schauder basis.
L-Limited and Almost L-Limited Sets in Dual Banach Lattices
Journal of Mathematical Extension, 2018
Following the concept of L-limited sets in dual Banach spaces introduced by Salimi and Moshtaghioun, we introduce the concept of almost L-limited sets in dual Banach lattices and then by a class of disjoint limited completely continuous operators on Banach lattices, we characterize Banach lattices in which almost L-limited subsets of their dual, coincide with L-limited sets.
L-Dunford–Pettis and Almost L-Dunford–Pettis Sets in Dual Banach Lattices
International Journal of Analysis and Applications, 2018
Following the concept of L-limited sets in dual Banach spaces introduced by Salimi and Moshtaghioun, we introduce the concepts of L-Dunford-Pettis and almost L-Dunford-Pettis sets in dual Banach lattices and then by a class of operators on Banach lattices, so called disjoint Dunford-Pettis completely continuous operators, we characterize Banach lattices in which almost L-Dunford-Pettis subsets of their dual, coincide with L-Dunford-Pettis sets. Also if A ⊆ X * and every weak null sequence (x n) in X converges uniformly on A, we say that A is an L-set. Every relatively compact subset of E is DP. If every DP subset of a Banach space X is relatively compact, then X has the relatively compact DP property (abb. DP rc P). For example, dual Banach spaces with the weak Radon-Nikodym property (see [11], in short W RN P) and Schur spaces (i.e., weak and norm
Duality, reflexivity and atomic decompositions in Banach spaces
Studia Mathematica, 2009
We study atomic decompositions and their relationship with duality and reflexivity of Banach spaces. To this end, we extend the concepts of "shrinking" and "boundedly complete" Schauder basis to the atomic decomposition framework. This allows us to answer a basic duality question: when an atomic decomposition for a Banach space generates, by duality, an atomic decomposition for its dual space. We also characterize the reflexivity of a Banach space in terms of properties of its atomic decompositions.
The Schur and (weak) Dunford-Pettis properties in Banach lattices
Journal of the Australian Mathematical Society, 2002
We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that l1, c0 and l∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class ∧. We also present examples of weighted Orlicz spaces with the Schur property which are not L1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.
Invariant subspaces of positive strictly singular operators on Banach lattices
Journal of Mathematical Analysis and Applications, 2008
It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators. introduction Read [Read91] presented an example of a strictly singular operator with no (closed non-zero proper) invariant subspaces. It remains an open question whether every positive strictly singular operator on a Banach lattice has an invariant subspace. The present paper contains several results in this direction.