Linear dynamical systems on Hilbert spaces : typical properties and explicit examples / S. Grivaux, É. Matheron, Q. Menet.

Format:

Book

Author/Creator:

Grivaux, S., author.

Matheron, Étienne, author.

Menet, Q., 1988- author.

Series:

Memoirs of the American Mathematical Society ; v. 1315.

Memoirs of the American Mathematical Society, 1947-6221 ; v. 1315

Language:

English

Subjects (All):

Hilbert space.

Linear systems.

Physical Description:

1 online resource (pages cm.)

Place of Publication:

Providence : American Mathematical Society, [2021]

System Details:

Mode of access : World Wide Web

text file

Summary:

"We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. (i) A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic. (ii) A typical upper-triangular operator with coefficients of modulus 1 on the diagonal is ergodic in the Gaussian sense, whereas a typical operator of the form "diagonal with coefficients of modulus 1 on the diagonal plus backward unilateral weighted shift" is ergodic but has only countably many unimodular eigenvalues; in particular, it is ergodic but not ergodic in the Gaussian sense. (iii) There exist Hilbert space operators which are chaotic and U-frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic but not ergodic, and Hilbert space operators which are chaotic and topologically mixing but not U-frequently hypercyclic. We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties"-- Provided by publisher.

Contents:

Chapter 1. Introduction Chapter 2. Typical properties of hypercyclic operators Chapter 3. Descriptive set-theoretic issues Chapter 4. Ergodicity for upper-triangular operators Chapter 5. Periodic points at the service of hypercyclicity Chapter 6. Operators of \cct and of \cpt Chapter 7. Explicit counterexamples Chapter 8. A few questions Short list of abbreviations

Notes:

Includes bibliographical references.

Electronic reproduction. Providence, Rhode Island : American Mathematical Society. 2021

Description based on print version record.

Other Format:

Print version: Grivaux, S., Linear dynamical systems on Hilbert spaces :

ISBN:

9781470464684

Access Restriction:

Restricted for use by site license.