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

## TL;DR

This paper explores the typical properties and explicit examples of linear operators on Hilbert spaces, revealing complex dynamical behaviors such as chaos, ergodicity, and hypercyclicity through theoretical and constructive methods.

## Contribution

It provides new insights into the generic dynamical properties of Hilbert space operators and constructs explicit examples illustrating these phenomena.

## Key findings

- Typical hypercyclic operators are not topologically mixing and have no eigenvalues.
- A typical upper-triangular operator is ergodic in the Gaussian sense.
- Existence of operators with complex combinations of chaotic, hypercyclic, and mixing properties.

## 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.