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Author:

Grivaux, S., author.

Title:

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

Publisher:

American Mathematical Society,

Copyright Date:

2021

Description:

v, 147 pages : illustrations ; 26 cm.

Subject:

Hilbert space.
Linear systems.
Hilbert space.
Linear systems.
Operator theory -- General theory of linear operators -- Cyclic vectors, hypercyclic and chaotic operators.
Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] -- Ergodic theory [See also 28Dxx] -- Measure-preserving transformations.
General topology {For the topology of manifolds of all dimensions, see 57Nxx} -- Spaces with richer structures -- Baire category, Baire spaces.
General topology {For the topology of manifolds of all dimensions, see 57Nxx} -- Connections with other structures, applications -- Descriptive set theory (topological aspects of Borel, analytic, proj.

Other Authors:

Matheron, Étienne, author.
Menet, Q., 1988- author.

Notes:

Includes bibliographical references (pages 145-147).

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.

Series:

Memoirs of the American Mathematical Society, 0065-9266 ; number 1315

ISBN:

1470446634
9781470446635

OCLC:

(OCoLC)1228929217

LCCN:

2021016944

Locations:

OVUX522 -- University of Iowa Libraries (Iowa City)