Unbounded order continuous operators on Riesz spaces

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz spaces. We show that the collection of all uo-continuous linear functionals on a Riesz space E is a band of \(E^\sim \).     

On Riesz spaces with b-property and strongly order bounded operators

A Riesz space E is said to have the b-property if each subset that is order bounded in the bidual remains to be order bounded in E. Properties of a Riesz space with the b-property, the relationship between the b-property and various classes of operators are studied.     

Some characterizations of Riesz spaces in the sense of strongly order bounded operators

Positivity - We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in $$E^{sim sim }$$ and F is a Dedekind complete Riesz...     

Statistical order convergence in Riesz spaces

In this paper, we introduce the concepts of statistical monotone convergence and statistical order convergence in a Riesz space, and establish some basic facts. We show that the statistical order convergence and the statistical convergence in norm need not be equivalent in a normed Riesz space. Finally, we introduce the statistical order boundedness of a sequence in a Riesz space.     

Unbounded order convergence in dual spaces

A net (_xα)$(x_{\alpha })$ in a vector lattice X   is said to be unbounded order convergent (or uo-convergent, for short) to x∈X$x\in X$ if the net (|_xα−x|∧y)$(|x_{\alpha }-x|\wedge y)$ converges to 0 in order for all y∈_X+$y\in X_{+}$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let X   be a Banach lattice. We prove that every norm bounded uo-convergent net in X^?$X^{?}$ is w^?$w^{?}$-convergent iff X   has order continuous norm, and that every w^?$w^{?}$-convergent net in X^?$X^{?}$ is uo-convergent iff X is atomic with order continuous norm. We also characterize among σ  -order complete Banach lattices the spaces in whose dual space every simultaneously uo- and w^?$w^{?}$-convergent sequence converges weakly/in norm.     

Riesz spaces of real continuous functions

An Archimedean Riesz space E is isomorphic to C(X) for some completely regular Hausdorff space X if and only if there exists a weak order unit e > 0 for which E is e-uniformly complete, e-semisimple, e-separating and 2-universally e-complete.     

Riesz transforms associated to Schrödinger operators on weighted Hardy spaces

Let w be some Ap weight and enjoy reverse Hölder inequality, and let L=−Δ+V be a Schrödinger operator on Rn, where is a non-negative function on Rn. In this article we introduce weighted Hardy spaces associated to L in terms of the area function characterization, and prove their atomic characters. We show that the Riesz transform ∇L−1/2 associated to L is bounded on for 1<p<2, and bounded from to the classical weighted Hardy space .     

Characterization of regular Riesz spaces using oru-compact operators

Dedicated to Professor Romulus Cristescu on the occasion of his 60-th birthday     

On discrete and continuous quotient Riesz spaces

Order properties of quotient Riesz spaces E/N(f) by null ideals N(f) are investigated. We show relationships between properties of a Riesz space E and its order dual E ^~ and properties of quotients E/N(f) where f runs over some subspaces of E ^~. A characterization of metrizable locally convex topological Riesz spaces whose all quotients (by proper closed ideals) are discrete is also given.     

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Invariant subspaces for positive operators on locally convex solid Riesz spaces

We prove that an x₀-quasinilpotent semigroup S of continuous positive linear operators on a locally convex solid Riesz space X has a common invariant subspace. Using this, a result which implies the main theorem of Abramovich, Aliprantis and Burkinshaw [J. Funct. Anal. 115 (1993) 418424] is also given.     

Representations of Archimedean Riesz spaces by continuous functions

A brief survey of representations of Archimedean Riesz spaces in spaces of continuous extended real-valued functions, together with an example of their use in proving results about Riesz spaces     

Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces

As a generalization of the notion of Riesz seminorm, a class of seminorms on directed partially ordered vector spaces is introduced, such that (1) every seminorm in the class can be extended to a Riesz seminorm on every larger Riesz space that is majorized and (2) a seminorm on an order dense linear subspace of a Riesz space is in the class if and only if it can be extended to a Riesz seminorm on the Riesz space. The latter property yields that if a directed partially ordered vector space has an appropriate Riesz completion, then a seminorm on the space is in the class if and only if it is the restriction of a Riesz seminorm on the Riesz completion. An explicit formula for the extension is given. The class of seminorms is described by means of a notion of solid unit ball in partially ordered vector spaces. Some more properties concerning restriction and extension are studied, including extension to L- and M-seminorms.     

Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of \(C_0(X)\), for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces.     

Banach-Saks operators on spaces of continuous functions

Summary Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm. We study in this paper Banach-Saks operators defined on C(K, E) spaces. We characterize these operators for a large class of compacts K (the scattered ones), or for a large class of Banach spaces E (the superreflexive ones). We also show by some examples that our theorems can not be extended directly.Partially supported by C.A.I.C.Y.T. grant 0338-84. The author wishes to thank Professor F.Bombal for his encouragement.     

A characterization of an order ideal in Riesz spaces

In this paper we give a characterization of order ideals in Riesz spaces.