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The Linear Dynamics of Several Commuting Operators
Nasca, Angelo J, III

2015, Doctor of Philosophy, Ohio State University, Mathematics.
Linear dynamics is the study of orbits of linear operators on Hilbert spaces (or more general topological vector spaces). If a linear operator has a dense orbit, we say the operator is hypercyclic. One of the fundamental results in linear dynamics is Ansari’s Theorem: If a bounded linear operator T is hypercyclic, then for any fixed n natural number, the operator T^n is also hypercyclic. The phenomenon exhibited by Ansari’s Theorem is a driving force behind much of the work in this document.

This thesis has an introductory chapter and two main chapters. The first main chapter has the goal of extending Ansari’s Theorem to actions generated by several commuting linear operators. This goal is realized under the assumption that the action in question is weakly mixing. This chapter also contains several results about semigroups of linear operators on finite dimensional spaces, including a multi-parameter analogue of Ansari’s Theorem, and ends with a treatment of affine hypercyclic operators.

The second main chapter extends the construction of a probability measure of full support on a Hilbert space which is preserved by a linear operator, to a situation involving the action generated by several commuting operators. Conditions are given that ensure mixing properties of such an action. Following this, we use joint ergodicity to establish some new theorems about joint hypercyclicity, including a theorem which is a variation on both Ansari’s Theorem and the related Leon-Muller Theorem. We end the chapter with an example of an operator which is jointly hypercyclic along the polynomial sequences n,n^2,...,n^{d-1}, and yet fails to be hypercyclic along the sequence n^d.
Vitaly Bergelson (Advisor)
136 p.

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