Generic convergence of infinite products of positive linear operators
type: Item
instance of: scholarly article
title: Generic convergence of infinite products of positive linear operators
zbMATH Open document ID: 0946.47023
DOI: 10.1007/BF01196385
author: Simeon Reich
author: Alexander J. Zaslavski
published in: Integral Equations and Operator Theory
publication date: 2000-10-17T00:00:00Z
review text: Let \(X\) be a Banach space ordered by a closed convex cone \(X_+\) with nonempty interior such that the norm is monotonic. Denote by \({\mathfrak M}\) the set of all sequences of monotonic linear operators. For \(\eta \in \text{int}(X_+)\) the interval \(\langle -\eta,\eta \rangle\) induces an equivalent norm on \(X\). This norm leads to two metrizable topologies (strong and weak) on various subspaces of \({\mathfrak M}\). The authors present several results about the asymptotic behavior of infinite products of generic elements of these subspaces. In addition to a weak ergodic theorem the authors also obtain convergence to a 1-dimensional operator \(f \otimes \eta\) where \(f\) is a continuous linear functional and \(\eta\) is a common fixed point.
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 1374991
zbMATH Keywords: ordered Banach spaces
zbMATH Keywords: monotonic linear operators
zbMATH Keywords: weak ergodic theorem
zbMATH Keywords: infinite products of operators
zbMATH Keywords: common fixed point
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
cites work: Q3748810
cites work: Q5518790
cites work: Q3843846
cites work: Ergodic theorems in demography
cites work: Q4098245
cites work: Generic flows generated by continuous vector fields in Banach spaces
cites work: Convergence of random products of compact contractions in Hilbert space
cites work: A generalization of a theorem of Amemiya and Ando on the convergence of random products of contractions in Hilbert space
cites work: Random Products of Contractions in Banach Spaces
cites work: Q4026288
cites work: Q5518786
cites work: Convergence of inhomogeneous products of matrices and coefficients of ergodicity
cites work: Q5786104
cites work: On infinite products of stochastic matrices
cites work: Unrestricted products of contractions in Banach spaces
cites work: Perron-Frobenius theory for Banach spaces with a hyperbolic cone
cites work: Q3035896
cites work: On rings of operators. Reduction theory
cites work: Convergence of generic infinite products of nonexpansive and uniformly continuous operators
cites work: Convergence of generic infinite products of homogeneous order-preserving mappings
cites work: Q4049667
cites work: Optimal Programs on Infinite Horizon 1
cites work: Dynamic properties of optimal solutions of variational problems
full work available at URL: https://doi.org/10.1007/bf01196385
OpenAlex ID: W2035808154
description: scientific article; zbMATH DE number 1374991