On positive strictly singular operators and domination

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.

Studia Mathematica, 2006

Given a positive Banach-Saks operator T between two Banach lattices E and F , we give sufficient conditions on E and F in order to ensure that every positive operator dominated by T is Banach-Saks. A counterexample is also given when these conditions are dropped. Moreover, we deduce a characterization of the Banach-Saks property in Banach lattices in terms of disjointness.

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.

Studia Math, 2008

It is proved that every positive Banach-Saks operator T : E → F between Banach lattices E and F factors through a Banach lattice with the Banach-Saks property, provided that F has order continuous norm. By means of an example we show that this order continuity condition cannot be removed. In addition, some domination results, in the Dodds-Fremlin sense, are obtained for the class of Banach-Saks operators.

2010

Quiero concluir esta introducción mostrando mi agradecimiento a todas aquellas personas que me han ayudado en la realización de este proyecto. No puedo sino comenzar por mis directores de Tesis: Francisco Hernández y Julio Flores, sin cuyo apoyo y estímulo esta Tesis habría sido imposible, pues con su paciencia y comprensión, me han guiado en estos años, compartiendo momentos mejores y peores. Por supuesto, va también mi agradecimiento a todos los miembros del Departamento de Análisis Matemático de la Universidad Complutense de Madrid que de uno u otro modo han contribuido a hacer realidad este trabajo. También quiero agradecer el buen recibimiento y trato recibido en el Department of Mathematics de la University of Missouri-Columbia y en el Equipe d'Analyse Fonctionnelle de la Université Paris VI/Paris VII.

Positivity, 2003

We prove that each positive operator from a Köthe function-space E(μ) to a Banach lattice F with a narrow majorant is itself narrow provided the norm on F is order continuous. We also prove that every l 2-strictly singular regular operator from L p[0,1], 1≤p F is narrow, provided F has an order continuous norm.

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.

2006

I herby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

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.