A quantitative approach to disjointly non-singular operators

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.

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.

Positivity

Given an operator T : X --> Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices TE : E(X) --> E(Y ) given by TE(f)(x) = T(f(x)). It is proved that for any Banach lattice E which does not contain c0, the operator T is an isomorphism on a subspace isomorphic to c0 if and only if so is TE. An analogous result for invertible operators on subspaces isomorphic to l1 is also given.

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.

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.

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).

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.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1996

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

Irish Mathematical Society Bulletin, 2008

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.