Linear dynamical systems on Hilbert spaces: typical properties and explicit examples

Abstract

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. - A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic. - A typical upper-triangular operator is ergodic in the Gaussian sense, whereas a typical operator of the form "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. - There exist Hilbert space operators which are chaotic and $\mathcal{U}$-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 $\mathcal{U}$-frequently hypercyclic. We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties.

124 pages

References in corpus (4)

• Recurrence properties of hypercyclic operators
• Strong mixing measures for linear operators and frequent hypercyclicity
• Category theorems for stable operators on Hilbert spaces
• Set theory and cyclic vectors

Cited by in corpus (16)

• Frequently recurrent operators
• Recurrent subspaces in Banach spaces
• Inverse of frequently hypercyclic operators
• Two remarks on the set of recurrent vectors
• On Shadowing and Chain Recurrence in Linear Dynamics
• On the complete metrisability of spaces of contractive semigroups
• The space of contractive $C_0$-semigroups is a Baire space
• Interpolation and non-dilatable families of ${\mathcal{C}}_0$-semigroups
• Dynamical properties for composition operators on $H^2({\mathbb{C}}_{+})$
• Inverse of $\mathcal{U}$-frequently hypercyclic operators
• Typicality of operators on Fréchet algebras admitting a hypercyclic algebra
• Local spectral properties of typical contractions on \(\ell_p,\)-spaces
• Trichotomy for the orbits of a hypercyclic operator on a Banach space
• Similar operator topologies on the space of positive contractions
• Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree
• Does a typical ${\ell }_p$-space contraction have a non-trivial invariant subspace?