Some properties of the space of regular operators on atomic Banach lattices (original) (raw)

Some characterizations of KB-operators on Banach lattices and ordered Banach spaces

Turkish Journal of Mathematics, 2020

We determine that two recent classes of KB-operators and weak KB-operators and the well-known class of b -weakly compact operators, from a Banach lattice into a Banach space, are the same. We extend our study to the ordered Banach space setting by showing that a weak chain-preserving operator between two ordered Banach spaces is a KB-operator if and only if it is a weak KB-operator.

Contribution to operators between Banach lattices

arXiv (Cornell University), 2022

In this paper we introduce and study a new class of operators related to norm bounded sets on Banach Lattice and which brings together several classical classes of operators (as oweakly compact operators, b-weakly compact operators, M-weakly compact operators, L-weakly compact operators, almost Dunford-Pettis operators). As consequences, we give some new lattice approximation properties of these classes of operators.

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.

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.

Characterizations of strictly singular operators on Banach lattices

Journal of the London …, 2009

New characterizations of strictly singular operators between Banach lattices are given. It is proved that for Banach lattices X and Y , such that X has fi nite cotype and Y satis es a lower 2-estimate, an operator T : X -->Y is strictly singular if and only if it is disjointly strictly singular and l2-singular. Moreover, if T is regular the same equivalence holds provided Y is just order continuous. Furthermore, it is shown that these results fail if the conditions on the lattices are relaxed.

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.

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.

Uniform Kadec-Klee Property In Banach Lattices

Arxiv preprint math/9406209, 1994

We prove that a Banach lattice X which does not contain the l n ∞uniformly has an equivalent norm which is uniformly Kadec-Klee for a natural topology τ on X. In case the Banach lattice is purely atomic, the topology τ is the coordinatewise convergence topology.