Using the notion of order convergent nets, we develop an order-theoretic approach to differentiable functions on Archimedean complex Φ-algebras. Most notably, we improve the Cauchy-Hadamard formulas ...for universally complete complex vector lattices given by both authors in a previous paper in order to prove that analytic functions are holomorphic in this abstract setting.
In this article we continue investigation of the lateral order on complex vector lattices started in 10,39. We extend some of main results of 35 and 32 to the setting of operators on complex vector ...lattices. We establish the Riesz-Kantorovich type calculus for regular orthogonally additive operators defined on the complexification EC of an uniformly complete vector lattice E with the principal projection property and taking values in Dedekind complete vector lattice F. We prove that a regular orthogonally additive operator T:EC→F from the complexfication EC of an uniformly complete vector lattice E with the principal projection property to a Banach lattice F with an order continuous norm is narrow if and only if the modulus |T|:EC→F is.
It was proved by Anthony Wickstead that the cone of positive linear operators between Banach lattices E and F coincides with the strongly closed convex hull of the set of lattice homomorphisms from E ...to F if and only if the cone of positive elements on E is the weakly closed convex hull of the union of extremal rays of the cone of positive liner functionals on E. This note aims to show that this result extends to a wider class of operators, namely, orthogonally additive homogeneous polynomials acting between vector lattices.
Suppose X is a vector lattice and there is a notion of convergence xα→σx in X. Then we can speak of an “unbounded” version of this convergence by saying that xα→uσx if |xα−x|∧u→σ0 for every u∈X+. In ...the literature, the unbounded versions of the norm, order and absolute weak convergence have been studied. Here we create a general theory of unbounded convergence but with a focus on uo-convergence and those convergences deriving from locally solid topologies. We will see that, not only do the majority of recent results on unbounded norm convergence generalize, but they do so effortlessly. Not only that, but the structure of unbounded topologies is clearer without a norm. We demonstrate this by removing metrizability, completeness, and local convexity from nearly all arguments, while at the same time making the proofs simpler and more general. We also give characterizations of minimal topologies in terms of unbounded topologies and uo-convergence.
It is proved that for a given two positive integers N,n with N<n and N orthoregular homogeneous polynomials of the same degree acting between Archimedean vector lattices and pairwise independent in ...some sense, the sum of these polynomials each raised to the power of n and multiplied by an orthomorphism, is orthogonally additive if and only if all these polynomials multiplied by the corresponding orthomorphisms are disjointness preserving. It is also shown that given three positive integers N,n,m, an order bounded orthogonally additive polynomial with values in a Dedekind complete vector lattice is the sum of n-th powers of N order bounded disjointness preserving m-homogeneous polynomials if and only if it is representable as a disjoint sum of N disjointness preserving mn-homogeneous polynomials.
A linear operator T between two lattice-normed spaces is said to be p-compact if, for any p-bounded net xα, the net Txα has a p-convergent subnet. p-Compact operators generalize several known classes ...of operators such as compact, weakly compact, order weakly compact, AM-compact operators, etc. Similar to M-weakly and L-weakly compact operators, we define p-M-weakly and p-L-weakly compact operators and study some of their properties. We also study up-continuous and up-compact operators between lattice-normed vector lattices.
Using new notions of consistent sets and levels in Riesz spaces, we study the relationship between linear and orthogonally additive operators on Banach lattices. We define a norm on the Riesz space ...U(E,F) of order bounded orthogonally additive operators acting between Banach lattices E,F and prove that if F is Dedekind complete then the set UB(E,F) of all bounded with respect to this norm elements of U(E,F) is a Dedekind complete Banach lattice which is a sublattice of U(E,F). One of the results asserts that if E is an AL-space, F a Banach lattice with a Fatou-Levi norm (in particular, a KB-space) then the Banach lattice L(E,F)=Lr(E,F) of all continuous linear (regular) operators from E to F is a 1-complemented subspace of UB(E,F). If, moreover, both E and F are separable then for every S∈L(E,F)+ there exists T∈UB(E,F) such that for every S1∈L(E,F) with |S1|≤S there is a contractive projection Φ of UB(E,F) onto L(E,F) with Φ(T)=S1. Note that, being a subspace of UB(E,F), the Banach lattice Lr(E,F) is not a sublattice of UB(E,F), because the order on Lr(E,F) differs from the order on UB(E,F).
We consider C-compact orthogonally additive operators in vector lattices. After providing some examples of C-compact orthogonally additive operators on a vector lattice with values in a Banach space ...we show that the set of those operators is a projection band in the Dedekind complete vector lattice of all regular orthogonally additive operators. In the second part of the article we introduce a new class of vector lattices, called C-complete, and show that any laterally-to-norm continuous C-compact orthogonally additive operator from a C-complete vector lattice to a Banach space is narrow, which generalizes a result of Pliev and Popov.
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