ON THE FUNCTIONAL CALCULUS IN ARCHIMEDEAN RIESZ SPACES WITH APPLICATIONS TO APPROXIMATION THEOREMS

ABSTRACT

We show that the functional calculus defined on the class of Dedekind σ-complete Riesz spaces can be extended to the class of uniformly complete Archimedean Riesz spaces without representing in the process the spaces involved by spaces of functions. As a consequence some results in the theory of Riesz spaces which were proved previously by representation techniques, can now be proved in an intrinsic way.     

COMPONENTS OF POSITIVE ELEMENTS IN RIESZ SPACES

Abstract

We find a necessary and sufficient condition on a Riesz space E such that a sub-Boolean algebra of components of a positive element generates the Boolean algebra of all components of the element. This condition yields a dual characterization of principal A-modules in the sense of D. Vuza. When applied to l-algebras, one finds improvements of results of J. Synnatzschke.     

AN f-ALGEBRA APPROACH TO THE RIESZ TENSOR PRODUCT OF ARCHIMEDEAN RIESZ SPACES

Abstract

We construct the Riesz tensor product of Archimedean Riesz spaces and derive its properties using functional calculus and f-algebras. We improve results on the approximation of elements in the Riesz tensor product by means of elements in the vector space tensor product in such a way that the order density property is a consequence of the improved approximation result.     

DUALITY BETWEEN CARLEMAN AND SEMI-CARLEMAN OPERATORS IN RIESZ SPACES

Abstract

The relation between Carleman and semi-Carleman operators is studied and it is shown that Carleman and semi- Carleman operators are mutually dual with respect to the formation of adjoints. This generalizes work of VB Korotkov.     

COMPLEMENTED RANGE SPACES

1. Abstract

Most of the well known perturbation results hold for linear operators whose nullity or defect is finite, which implies the existence of topological complements to the null and range spaces respectively. This paper investigates properties associated with the stability of the complementability of the range space under various perturbations.     

A MULTIPLICATION INEQUALITY IN COMPLEX BANACH LATTICE ALGEBRAS

Abstract

We present an alternative order theoretic proof for the fundamental result that |Z₁, Z₂ | ? |Z₁ 1 | Z₂ | with z₂ and z₂ elements of the complexification of an Archimedean uniformly complete Riesz algebra.     

H-ULTRABORNOLOGICAL SPACES

Abstract

In this paper we define a special type of H-locally convex space—H-ultrabornological space -. We shall characterize those H-locally convex spaces in terms of linear maps and of inductive limits.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

The Root Problem For Stochastic Operators

Abstract

This note contains an order-theoretical approach to the theory of stochastic operators. Our main result is a necessary condition for the existence of n-th roots of an ergodic and conservative operator, which generalizes known results for measure-preserving operators and operators on L ^p-type spaces.     

ON STRICT SINGULARITY OF LOCALLY INTEGRAL OPERATORS ON BANACH FUNCTION SPACES

Abstract

In this note we generalize a result of L. Weis and V. V. Shevchik about strict singularity of the Fourier transform to a large class of locally integral operators on function spaces.     

SOME ASPECTS OF THE THEORY OF SYMMETRIC OPERATOR SPACES

Abstract

We discuss several aspects of the theory of symmetric Banach spaces of measurable operators, including their construction and certain topological and geometric properties. Particular emphasis is given to the role played by rearrangement inequalities.     

On von Neumann regular f-algebras

It is shown that for a large class of f-algebras, von Neumann regularity and -lateral completeness are equivalent notions.     

DENTABILITY IN LOCALLY CONVEX SPACES

Abstract

Let A be a non-empty bounded subset of a locally convex space E. We show that if all the separable subsets of A are weakly metrisable, then the weak*-compact subsets of E¹ satisfy geometrical conditions which are similar to the concept of “dentability” used to characterise the Radon-Nikodý Property in dual Banach spaces.     

A RADON-NIKODYM THEOREM FOR RIESZ SPACE VALUED MEASURES

ABSTRACT

In this note, as a consequence of the Radon-Nikodym theorem for positive operators given by Luxemburg and Schep ([6]), we obtain a Radon-Nikodym theorem for Riesz space valued measures similar to Wright's theorem ([11]).     

ON TOPOLOGICAL LINEAR CONTRACTIONS ON NORMED SPACES AND APPLICATION

51.IntroductionandPreliminariesThroughoutthispaper(X,11'if)denotesanormedspaceandN(11'II)denotesthecollectionofallnormsonXwhichareequivalentto11'11.Foreach11'if'6N(II'II)andeachr>0,letX.(ll'II')~{xEX:Ilxll'5r}andX?(ll'if')~{xEX.Ilxll'相似文献   

THE WEAK RADON-NIKODYM PROPERTY IN SPACES OF NUCLEAR OPERATORS

Abstract

We prove that if X and Y are Banach spaces such that X* has the weak Radon-Nikodym property (WRNP), Y has the Radon-Nikodym property (RNP) and Y is complemented in its bidual, then the space N(X,Y) of all nuclear operators from X to Y has the WRNP. If moreover X* has the RNP, then N(X,Y) has the RNP.     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.     

CONSTRAINED CONTROLLABILITY IN NON REFLEXIVE BANACH SPACES

Abstract

Throughout the literature on optimal control in Banach spaces, hypotheses like “separable and reflexive” are frequently encountered. In this note we consider one such case, studied by Peichl and Schappacher. Using techniques from Banach space theory and the theory of vector measures, we show how to remove the hypothesis of reflexivity and translate the problem of controls to one about the strong continuity of an adjoint semigroup on the positive real axis.     

Weyl’s Theorem and Perturbations

In the present paper we examine the stability of Weyl’s theorem under perturbations. We show that if T is an isoloid operator on a Banach space, that satisfies Weyl’s theorem, and F is a bounded operator that commutes with T and for which there exists a positive integer n such that Fⁿ is finite rank, then T + F obeys Weyl’s theorem. Further, we establish that if T is finite-isoloid, then Weyl’s theorem is transmitted from T to T + R, for every Riesz operator R commuting with T. Also, we consider an important class of operators that satisfy Weyl’s theorem, and we give a more general perturbation results for this class.