38 research outputs found
On KB and Levi operators in Banach lattices
We prove that an order continuous Banach lattice E is a KB-space if and only
if each positive compact operator on E is a KB operator. We give conditions on
quasi-KB (resp., quasi-Levi) operators to be KB (resp., Levi), study norm
completeness and domination for these operators, and show that neither KB nor
Levi operators are stable under rank one perturbations
Full Lattice Convergence on Riesz Spaces
The full lattice convergence on a locally solid Riesz space is an abstraction
of the topological, order, and relatively uniform convergences. We investigate
four modifications of a full convergence on a Riesz space. The
first one produces a sequential convergence . The second makes an
absolute -convergence and generalizes the absolute weak
convergence. The third modification makes an unbounded -convergence
and generalizes various unbounded convergences recently studied in the
literature. The last one is applicable whenever is a full
convergence on a commutative -algebra and produces the multiplicative
modification of . We study general properties of full
lattice convergence with emphasis on universally complete Riesz spaces and on
Archimedean -algebras. The technique and results in this paper unify and
extend those which were developed and obtained in recent literature on
unbounded convergences
Limitedly L-weakly compact operators
We introduce new class of limitedly L-weakly compact operators from a Banach
space to a Banach lattice. This class is a proper subclass of the
Bourgain-Diestel operators and it contains properly the class of L-weakly
compact operators. We give its efficient characterization in term of sequences,
investigate the domination problem, and study the completeness of this class of
operators
Duality and Norm Completeness in the Classes of Limitedly--Lwc and Dunford--Pettis--Lwc Operators
We investigate the duality and norm completeness in the classes of
limitedly--L-weakly compact and Dunford--Pettis--L-weakly compact and operators
from Banach spaces to Banach lattices
Unbounded p-Convergence in Lattice-Normed Vector Lattices
A net xα in a lattice-normed vector lattice (X, p, E) is unbounded p-convergent to x ∈ X if p(| xα− x| ∧ u) → o 0 for every u ∈ X+. This convergence has been investigated recently for (X, p, E) = (X, |·|, X) under the name of uo-convergence, for (X, p, E) = (X, ‖·‖, ℝ) under the name of un-convergence, and also for (X, p, ℝX ′) , where p(x)[f]:= |f|(|x|), under the name uaw-convergence. In this paper we study general properties of the unbounded p-convergence.Article Pre-prin