## Operators Grow on Trees

- Nov. 15, 2019
- 2:30 p.m.

## Abstract

The study of rooted trees, both finite and infinite, is an elegant topic with wide applications. While tree-theory is traditionally in the jurisdiction of discrete mathematicians, there are certainly applications to other areas of mathematics as well. In this discussion, we'll denote by l^2 (T) the space of all square-summable labelings on a given rooted tree, T. As long as certain conditions are met, we can define a Tree-Shift Operator that is a generalization of the classic Shift Operator that maps from l^2 to itself. In this manner, each rooted tree has a Tree-Shift Operator associated to it. After seeing a few examples, we will ask whether or not rooted trees are completely determined by their associated operator. In other words, If a pair of trees enjoys the property that their Tree-Shift Operators are the same (up to unitary equivalence), can we say that the underlying trees are isomorphic? We will investigate this question in the finite tree case and the infinite tree case separately, and see some results and some open problems.