SOME PROPERTIES OF STRICTLY 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.

Quarterly Journal of Mathematics, 2007

It is proved that every positive operator R on a Banach lattice E dominated by a strictly singular operator T : E → E satisfies that the R 4 is strictly singular. Moreover, if E is order continuous then the R 2 is already strictly singular.

Arkiv för matematik, 1978

2000

We prove that each positive operator from a Banach lattice E to a Banach lattice F with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on F is order continuous. We prove as well that if S : E → E is dominated by a disjointly strictly singular operator, then S 2 is disjointly strictly singular.

Positivity, 2015

In this paper we give several new results concerning domination problem in the setting of positive operators between Banach lattices. Mainly, it is proved that every positive operator R on a Banach lattice E dominated by an almost weakly compact operator T satisfies that the R 2 is almost weakly compact. Domination by strictly singular operators is also considered. Moreover, we present some interesting connections between strictly singular, disjointly strictly singular and almost weakly compact operators.

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.

2019

A Banach lattice algebra is a Banach lattice, an associative algebra with a sub-multiplicative norm and the product of positive elements should be positive. In this note we study the Arens regularity and cohomological properties of Banach lattice algebras.

Mathematische Annalen, 1992

Using compactness properties of bounded subsets of spaces of vector measure integrable functions and a representation theorem for q-convex Banach lattices, we prove a domination theorem for operators between Banach lattices. We generalize in this way several classical factorization results for operators between these spaces, as psumming operators.

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.

2006

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Spectral properties of strictly singular and disjointly strictly singular operators on Banach lattices are studied. We show that even in the case of positive operators, the whole spectral theory of strictly singular operators cannot be extended to disjointly strictly singular. However, several spectral properties of disjointly strictly singular operators are given.