Some loose ends on unbounded order convergence

The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.     

Continuous functionals for unbounded convergence in Banach lattices

Archiv der Mathematik - Recently, the different types of unbounded convergences $$(uo, un, uaw, ua w^*)$$ in Banach lattices were studied. In this paper, we study the continuous functionals with...     

Duality for unbounded order convergence and applications

Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) \({X_{uo}^\sim }\) of a Banach lattice X and identify it as the order continuous part of the order continuous dual \({X_n^\sim }\). The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the Fenchel–Moreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance.     

Unbounded norm convergence in Banach lattices

A net \((x_\alpha )\) in a vector lattice X is unbounded order convergent to \(x \in X\) if \(|x_\alpha - x| \wedge u\) converges to 0 in order for all \(u\in X_+\). This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net \((x_\alpha )\) in a Banach lattice X is unbounded norm convergent to x if Open image in new window for all \(u\in X_+\). We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.     

Unbounded asymptotic equivalences of operator nets with applications

Positivity - Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two...     

Some Applications of Rademacher Sequences in Banach Lattices

We give several applications of Rademacher sequences in abstract Banach lattices. We characterise those Banach lattices with an atomic dual in terms of weak* sequential convergence. We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the compactness of operators from L^p into L^q. Finally we generalise earlier work of ours by showing that, amongst Banach lattices F with an order continuous norm, those having the property that the linear span of the positive compact operators fromE into F is complete under the regular norm for all Banach lattices E are precisely the atomic lattices.     

Invariant sublattices for positive operators

There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces.     

On the notion of stability of order convergence in vector lattices

In the theory of Banach lattices the following criterion for a norm to be order continuous is established: a norm is order continuous if and only if every order bounded sequence of positive pairwise disjoint elements in a lattice converges to zero in norm. In this paper we give a criterion for order convergence to be stable in a rather wide class of vector lattices which includes all Köthe spaces. The formulation of the criterion is analogous to that of the above-mentioned criterion for a norm to be order continuous. Namely, under certain conditions imposed on a vector lattice, stability of order convergence is equivalent to the condition that every order bounded sequence of positive pairwise disjoint elements converges relatively uniformly to zero. Furthermore, we study some types of order ideals in vector lattices. In terms of these ideals we give clarified versions of the above-stated criterions. As for notation and terminology, see for example [1,2].Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1026–1031, September–October, 1994.     

High order splitting methods for analytic semigroups exist

In this paper, we are concerned with the construction and analysis of high order exponential splitting methods for the time integration of abstract evolution equations which are evolved by analytic semigroups. We derive a new class of splitting methods of orders three to fourteen based on complex coefficients. An optimal convergence analysis is presented for the methods when applied to equations on Banach spaces with unbounded vector fields. These results resolve the open question whether there exist splitting schemes with convergence rates greater then two in the context of semigroups. As a concrete application we consider parabolic equations and their dimension splittings. The sharpness of our theoretical error bounds is further illustrated by numerical experiments.     

Su talune disequazioni variazionali ellittiche non lineari del secondo ordine

Summary In this note, making use of a result of J. L. Lions, we examine some non linear elliptic variational inequalities defined on domains which may be unbounded. Such variational inequalities are associated to a uniformely second order elliptic operator. We start with the derivation of an existence theorem (on bounded domains) under non coerciveness assumptions. Next we examine the convergence for the solutions of a collection of variational inequalities. To this purpose we study convergence theorems for variational inequalities associated to operators belonging to a class of abstract mapping of pseudomonotone type between Banach spaces. The solvability of some variational inequalities on unbounded domains then follows directly.

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Entrata in Redazione il 6 aprile 1977.

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Lavoro eseguito nell'ambito del C.N.R., Laboratorio per la Matematica Applicata via L. B. Alberti 4, Genova.     

Some results on unbounded absolute weak Dunford–Pettis operators

We characterize Banach lattices for which each Dunford–Pettis operator (or weak Dunford–Pettis) is unbounded absolute weak Dunford–Pettis and conversely.

     

A characterization of dual Banach lattices

In this paper we give a characterization of dual Banach lattices. In fact, we prove that a Banach function space E on a separable measure space which has the Fatou property is a dual Banach lattice if and only if all positive operators from L¹(0,1) into E are abstract kernel operators, hence extending the fact, proved by M. Talagrand, that separable Banach lattices with the Radon-Nikodym property are dual Banach lattices.     

A Note on Weakly Isotone Maps and Common Solutions for Differential Systems

In this short note we examine the connection between weakly isotone maps and common solutions for first order Cauchy problems in R^n and, as a rule, in Banach lattices.     

On order polynomially complete lattices

One of the problems in the theory of order polynomially complete lattices is the question whether an order polynomially complete lattice is necessarily finite. In this note we give a partial answer to this problem by showing: No unbounded lattice is order polynomially complete. From this we deduce that a polynomially complete lattice cannot be countably infinite.Presented by I. Rosenberg.     

Separation properties for Carleman operators on Banach Lattices

Among the Carleman operators on Banach lattices, we consider the operators that are decomposition operators or lattice homormphisms. Using a generalization of an atom, characterizations for these two classes of operators on Banach lattices with order units are provided.     

Almost Dunford–Pettis sets in Banach lattices

We introduce and study the class of almost Dunford–Pettis sets in Banach lattices. It also discusses some of the consequences derived from this study. As an application, we characterize Banach lattices whose relatively weakly compact sets are almost Dunford–Pettis sets. Also, we establish some necessary and sufficient conditions on which an almost Dunford–Pettis set is L-weakly compact (respectively, relatively weakly compact). In particular, we characterize Banach lattices under which almost Dunford–Pettis sets in the topological dual of a Banach lattice coincide with that of L-weakly compact (respectively, relatively weakly compact) sets. As a consequences we derive some results.     

The penalty method for variational ineqalities with nonsmooth unbounded operators in banach space

In this paper we study the existence of a solution, convergence and stability of the penalty method for variational inequalities with nonsmooth unbounded uniformly and properly monotone operators in Banach space. All the objects of the inequality-the operator, “the data” and the set of constraints-are to be perturbed. The stability theorems are formulated in terms of geometric characteristics of the space and its dual. The results of this paper extend and generalize results of Lions [13]. They are new even in Hilbert spaces.     

On the convergence of constrained minima

We characterize the convergence of values and Lagrange multipliers for minimum problems of perturbed quadratic functionals in Banach spaces subject to a fixed linear operator constraint with finite-dimensional range. We obtain sufficient conditions for the convergence of the solutions. Examples show the different behaviour of the continuous dependence problem for the analogous unconstrained minimizations, where the gamma-type variational convergences are relevant. Such convergence conditions are extended here to the con strained problems.     

一阶导数条件下带参数的修正型Euler-Halley迭代族的收敛性

本文研究了Banach空间中非线性算子方程的带参数的修正型Euler-Halley迭代族的收敛性问题.在算子的一阶导数满足Lipschitz条件下建立了修正型Euler-Halley迭代族的半局部二阶收敛性.     

On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

In the paper we study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive.