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# Introduction

Published online by Cambridge University Press:  10 December 2009

## 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.

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Publisher: Cambridge University Press
Print publication year: 2009

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• Introduction
• Book: Dynamics of Linear Operators
• Online publication: 10 December 2009
• Chapter DOI: https://doi.org/10.1017/CBO9780511581113.001
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• Introduction
• 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
• Book: Dynamics of Linear Operators
• Online publication: 10 December 2009
• Chapter DOI: https://doi.org/10.1017/CBO9780511581113.001
Available formats
×