CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

In this talk we will be concerned with linear dynamics, an area of operator theory mainly concerned with the behavior of the iterates of linear transformations. One particular seductive feature of linear dynamics is the diversity of ideas and techniques that are involved in its study, owing to its strong connection with a number of distinct areas of mathematics. For some of them, e.g. combinatorics, number theory, topology, approximation theory and probability.

This is the first of a series of two talks. In this first talk, we overview some basic and striking facts (listed below) concerning the main object of study in linear dynamics: hypercyclic operators.

1. Hypercyclicity is a purely infinite-dimensional phenomenon: no finite dimensional space supports any hypercyclic operator;
2. It is not easy at all to determine whether a linear operator is hypercyclic. However, the set of hypercyclic operators is dense for the Strong Operator Topology in the algebra of linear and bounded operators;
3. Hypercyclicity is far from being an exotic phenomenon: any infinite-dimensional separable Frechet space supports a hypercyclic operator.

In the second talk we will see an example of the connection between linear dynamics and combinatorics.