Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-swqlm Total loading time: 0.342 Render date: 2021-11-30T05:33:59.775Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Introduction

Published online by Cambridge University Press:  10 December 2009

Frédéric Bayart
Affiliation:
Université de Clermont-Ferrand II (Université Blaise Pascal), France
Étienne Matheron
Affiliation:
Université d'Artois, France
Get access

Summary

Linear dynamics is a young and rapidly evolving branch of functional analysis, which was probably born in 1982 with the Toronto Ph.D. thesis of C. Kitai [158]. It has become rather popular, thanks to the efforts of many mathematicians. In particular, the seminal paper [123] by G. Godefroy and J. H. Shapiro, the authoritative survey [133] by K.-G. Grosse-Erdmann and the beautiful notes [222] by J. H. Shapiro have had a considerable influence on both its internal development and its diffusion within the mathematical community. After more than two decades of active research, this would seem to be the proper time to write a book about it.

As the name indicates, linear dynamics is mainly concerned with the behaviour of iterates of linear transformations. On finite-dimensional spaces, things are rather well understood since linear transformations are completely described by their Jordan canonical form. However, a new phenomenon appears in an infinite-dimensional setting: linear operators may have dense orbits. In fact, quite a lot of natural operators have this property.

To settle some terminology, let us recall that if T is a continuous linear operator acting on some topological vector space X, the T-orbit of a vector xX is the set O(x, T) := {x, T(x), T2(x), … }. The operator T is said to be hypercyclic if there exists some vector xX whose T-orbit is dense in X. Such a vector x is said to be hypercyclic for T.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Send book to Kindle

To send this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Introduction
  • Frédéric Bayart, Université de Clermont-Ferrand II (Université Blaise Pascal), France, Étienne Matheron
  • Book: Dynamics of Linear Operators
  • Online publication: 10 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511581113.001
Available formats
×

Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

  • Introduction
  • Frédéric Bayart, Université de Clermont-Ferrand II (Université Blaise Pascal), France, Étienne Matheron
  • Book: Dynamics of Linear Operators
  • Online publication: 10 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511581113.001
Available formats
×

Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

  • Introduction
  • Frédéric Bayart, Université de Clermont-Ferrand II (Université Blaise Pascal), France, Étienne Matheron
  • Book: Dynamics of Linear Operators
  • Online publication: 10 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511581113.001
Available formats
×