Asymptotically commuting families of operators

We study families of symmetric operators (Q _n) with domains given by the range of self-adjoint contraction semigroups (e ^−tHn ). Assuming the asymptotic commutativity, lim _n [Q _n, e^−tHn]=0, and certain other estimates, we establish the existence and properties of a limiting self-adjoint operatorQ=lim _n Q _n. We apply these results to the study of an elementary supersymmetry algebra. Supported in part by the National Science Foundation under Grant DMS/PHY 88-16214.     

Modal positive operators

For every modal positive operator, there exists a formula defining a least fixed point for that operator in partially ordered Kripke models with the property of being cofinal for every infinite ascending chain. A similar result obtains also for strict partially ordered Kripke models with the same property. Supported by the Russian Humanitarian Science Foundation (RHSF), grant No. 97-03-04089, and by the Siberian Branch of the Russian Academy of Science through PSO RAN No. 473, grant No. 3. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 585–597, September–October, 1999.     

Products of positive operators

A new, very simple proof is given of a result of P. Y. Wu which asserts that every unitary operator on an infinite-dimensional Hilbert space is a product of positive operators.

     

Interpolation of positive operators

We study problems of interpolation of positive linear operators in couples of ordered Banach spaces. From this viewpoint, we study couples of noncommutative spaces L ₁, L _∞ associated with weights and traces on von Neumann algebras.     

Certain positive linear operators

Properties (including the approximating ones) are investigated of positive linear operators L_n(f; x) for which the relation $$L_n \left( {\left( {t - x} \right)f\left( t \right); x} \right) = \frac{{\varphi \left( x \right)}}{n}L'_n \left( {f\left( t \right); x} \right)$$ is fulfilled, as well as the properties of operators L_n ^(m)(f;x). The results are applicable, in particular, to Bernstein polynomials, to the operators of Mirak'yan, Baskakov, and others.     

On positive transitive operators

In this short paper we prove that even though the latest transitive operator has only one negative entry, we cannot get rid of it by simple operations. The research is motivated by, still unsolved, invariant subspace problem for positive operators.      

Lebesgue-type decomposition of positive operators

The present note is a revision of Ando’s work with the same title. We give a new construction for the Lebesgue-decomposition of positive operators on Hilbert spaces with respect to each other. Our approach is similar to that of Kosaki: we use unbounded operator techniques and factorizations via two auxiliary Hilbert spaces associated to the positive operators in question.     

Rank factorizations of positive operators

The notion of a rank factorization of a positive operator through a cone is introduced and related to the nonnegative rank factorization of a nonnegative matrix. The concept is applied to the study of group inverses of certain positive operators.     

Factorisation of positive definite operators

In this paper we prove Reade’s result for the positive definite C ¹ kernels by using the factorisation method used by Kühn. Received: 10 April 2007, Revised: 3 December 2007     

Periodic points of positive linear operators and Perron-Frobenius operators

LetC(S) denote the Banach space of continuous, real-valued mapsf:S and letA denote a positive linear map ofC(S) into itself. We give necessary conditions that the operatorA have a strictly positive periodic point of minimal periodm. Under mild compactness conditions on the operatorA, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal periodm. We study a class of Perron-Frobenius operators defined by

On positive linear interpolation operators

В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x ₁ ⁽ⁿ⁾ /(n)<... _n ⁽ⁿ⁾ <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx _k ⁽ⁿ⁾ /(n) являются ну лями многочлена ЛагерраL _n ^(α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL _n ^(α) (x) при — 1相似文献   

Polyhedral cones and positive operators

Some results are obtained relating topological properties of polyhedral cones to algebraic properties of matrices whose columns are the extremal vectors of the cone. In addition, several characterizations of positive operators on polyhedral cones are given.