The set of periods of periodic points of a linear operator
type: Item
instance of: scholarly article
title: The set of periods of periodic points of a linear operator
zbMATH Open document ID: 1172.15002
author: J. Martínez
published in: Linear Algebra and its Applications
publication date: 2009-06-04T00:00:00Z
review text: For each \(n \in {\mathbb N}\) define the set \[ F_n=\{{\{1\} \cup \tilde A:A \subset {\mathbb N}\text{ and }|A|\leq n/2} \}\cup \{{\{1\}\cup \tilde A:A \subset {\mathbb N}\backslash \{1\},\;2 \in A\text{ and }|A|= (n+1)/2}\}, \] where \(| A |\) denotes the cardinality of \(A\) and \(\tilde A\) denotes the smallest subset of \({\mathbb N}\) containing \(A\) and closed under the binary operation least common multiple. Note also that \(F_{2m}= \{\{1\}\cup \tilde A:A \subset{\mathbb N}\) and \(|A|\leq m\}\) for all \(m \in {\mathbb N}\). The main results of this note are the following: (i) \(F_{2n}\) is the family of period sets of linear operators on \({\mathbb C}\); (ii) \(F_n\) is the family of period sets of linear operators on \({\mathbb R}\); (iii) \(F = \bigcup_n F_n\) is the family of period sets of linear operators on the Hilbert space \(l^2\) having finite rank (finite dimensional range); (iv) \(\{ {A \subset {\mathbb N}:A = \tilde A\text{ and }1 \in A} \}\) is the family of period sets of isometries on \(l^2\).
reviewed by: Mihail Voicu
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 5561624
zbMATH Keywords: linear operators
zbMATH Keywords: decomposition theorem
zbMATH Keywords: period sets
zbMATH Keywords: linear operators on sequence space
zbMATH Keywords: Hilbert space \(l^2\)
zbMATH Keywords: finite rank
zbMATH Keywords: finite dimensional range
MaRDI profile type: Publication
full work available at URL: j.laa.2009.02.027
OpenAlex ID: W1977255327
cites work: Fixpoints of Polynomials and Rational Functions
cites work: Dynamics in one dimension
cites work: Q3993502
cites work: Q3052960
cites work: Q5667036
DOI: 10.1016/J.LAA.2009.02.027
description: scientific article; zbMATH DE number 5561624