Disjointly homogeneous Banach lattices and compact products of operators (original) (raw)
Related papers
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.
Annals of Functional Analysis, 2018
We study several properties of the modulus of order bounded disjointness-preserving operators. We show that, if T is an order bounded disjointness-preserving operator, then T and |T | have the same compactness property for several types of compactness. Finally, we characterize Banach lattices having b-AM-compact (resp., AM-compact) operators defined between them as having a modulus that is b-AM-compact (resp., AM-compact).
Strictly singular and power-compact operators on Banach lattices
preprint
Compactness of the iterates of strictly singular operators on Banach lattices is analyzed. We provide suitable conditions on the behavior of disjoint sequences in a Banach lattice, for strictly singular operators to be Dunford-Pettis, compact or have compact square. Special emphasis is given to the class of rearrangement invariant function spaces (in particular, Orlicz and Lorentz spaces). Moreover, examples of rearrangement invariant function spaces of xed arbitrary indices with strictly singular non power-compact operators are also presented.
Disjointly homogeneous Banach lattices: Duality and complementation
Journal of Functional Analysis, 2014
We study several properties of disjointly homogeneous Banach lattices with a special focus on two questions: the self-duality of this class and the existence of disjoint sequences spanning complemented subspaces. Positive results concerning these problems are provided. Moreover, we give examples of reflexive disjointly homogenous spaces whose dual spaces are not, answering the first question in the negative.
Some results on AM-compact operators
Bulletin of the Belgian Mathematical Society - Simon Stevin
We characterize Banach lattices under which each AM-compact (resp. b-AM-compact) operator is Dunford-Pettis. Also, we study the AM-compactness of limited completely continuous operators.
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.
Compact groups of positive operators on Banach lattices
Indagationes Mathematicae, 2014
In this paper we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.
Some properties of the space of regular operators on atomic Banach lattices
Collectanea mathematica, 2011
Let L r (E, X ) denote the space of regular linear operators from a Banach lattice E to a Banach lattice X . In this paper, we show that if E is a separable atomic Banach lattice, then L r (E, X ) is reflexive if and only if both E and X are reflexive and each positive linear operator from E to X is compact; moreover, if E is a separable atomic Banach lattice such that E and E * are order continuous, then L r (E, X ) has the Radon-Nikodym property (respectively, is a KB-space) if and only if X has the Radon-Nikodym property (respectively, is a KB-space) and each positive linear operator from E to X is compact.
SOME PROPERTIES OF STRICTLY SINGULAR OPERATORS ON BANACH LATTICES
maia.ub.es
Several results obtained during the author's Ph.D. Thesis are presented. In particular, domination results (in Dodds-Fremlin sense) for the ideal of strictly singular operators will be given. Moreover, the connections between strictly singular and the classes of AM-compact, l2-singular and disjointly strictly singular are studied. As an application we obtain existence of invariant subspaces for positive strictly singular operators. On a di erent direction, results on compact powers of strictly singular operators are also presented extending a theorem of V. Milman. Finally, we study when a c0-singular or l1-singular operator can be extended to an operator between vector valued lattices preserving its singularity properties.
Invariant subspaces of positive strictly singular operators on Banach lattices
Journal of Mathematical Analysis and …, 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.