Recent Trends on Order Bounded Disjointness Preserving Operators

1998

We construct an invertible disjointness preserving operator T on a normed lattice such that T −1 is not disjointness preserving.

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).

Hacettepe Journal of Mathematics and Statistics, 2020

In this short note, our aim is to solve two problems in the theory of disjointness preserving operators. Firstly, we obtain the converse direction of Hart's Theorem which was given in [D.R. Hart, Some properties of disjointness preserving operators, Mathematics Proceedings, 1985]. As a result, we get an affirmative solution of an open problem given by Y.A. Abramovich and A.K. Kitover in [Inverses of disjointness preserving operators, Mem. Amer. Math. Soc., 2000].

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021

We introduce and study some operational quantities which characterize the disjointly non-singular operators from a Banach lattice E to a Banach space Y when E is order continuous, and some other quantities which characterize the disjointly strictly singular operators for arbitrary E.

Electronic Research Announcements of the American Mathematical Society, 2003

An order bounded disjointness preserving operator T T on an Archimedean vector lattice is algebraic if and only if the restriction of T n ! T^{n!} to the vector sublattice generated by the range of T m T^{m} is strongly diagonal, where n n is the degree of the minimal polynomial of T T and m m is its ‘valuation’.

Proceedings of the American Mathematical Society, 2003

We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded disjointness preserving operator. In this way, we constructively generalize results by Abramovich, Arensen and Kitover (1992), Grobler and Huijsmans (1997), Hart (1985), Kutateladze, and Meyer-Nieberg (1991).

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.

Canadian Journal of Mathematics, 2016

We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an "almost disjointness preserving" operator is well approximable by a disjointess preserving one. However, this approximation is not always possible, as our examples show.

Journal of Soviet Mathematics, 1991

The survey is devoted to the presentation of the state of the art of a series of directions of the theory of order-bounded operators in vector lattices and in spaces of measurable functions. The theory of disjoint operators, the generalized Hewitt-Yosida theorem, the connection with p-absolutely summing operators are considered in detail.

In this note we give an alternative proof of the Riesz-Kantorovich formula for the modulus of an order bounded disjointness preserving operator on complex vector lattices. Our approach is intrinsic and constructive. Also, we don't assume the vector lattices under consideration to be uniformly complete.