Riesz Inequality for the System of Root Functions of Second Order Ordinary Differential Operator

An ordinary differential operator of second order with coefficients is considered. The Riesz property of the system of root functions of the given operator is studied. The criterion of Bessel property in 2 L , of root functions system is established and use it to obtain sufficient conditions for the Riesz property of a system of normalized root functions of this operator in p L .

A Reverse Theorem on the·-w*Continuity of the Dual Map

Given a Banach spaceX,x∈𝖲X, and𝖩Xx=x*∈𝖲X*:x*x=1, we define the set𝖩X*xof allx*∈𝖲X*for which there exist two sequencesxnn∈N⊆𝖲X∖{x}andxn*n∈N⊆𝖲X*such thatxnn∈Nconverges tox,xn*n∈Nhas a subnetw*-convergent tox*, andxn*xn=1for alln∈N. We prove that ifXis separable and reflexive andX*enjoys the Radon-Riesz property, then𝖩X*xis contained in the boundary of𝖩Xxrelative to𝖲X*. We also show that ifXis infinite dimensional and separable, then there exists an equivalent norm onXsuch that the interior of𝖩Xxrelative to𝖲X*is contained in𝖩X*x.

Infinitary lattice and Riesz properties of pseudoeffect algebras and po-groups

Pseudoeffect (PE-) algebras generalize effect algebras by no longer being necessarily commutative. They are in certain cases representable as the unit interval of a unital po-group, for instance if they fulfil a certain Riesz property.Several infinitary lattice properties and the countable Riesz interpolation property are studied for PE-algebras on the one hand and for po-groups on the other hand. We establish the exact relationships between the various conditions that are taken into account, and in particular, we examine how properties of a PE-algebra are related to the analogous properties of a representing po-group.