Abstract:
  Given an analytic $\phi: \mathbb{D}\to \mathbb{D}$ fixing 0, Schroeder's
equation seeks an analytic function $f:\mathbb{D}\to \mathbb{C}$ satisfying
$f\circ \phi=\phi'(0)f$.  In 2003 Cowen and MacCluer generalized this
problem to the domain $\mathbb{B}^n=\{z \in \mathbb{C}^n; |z|<1\}$ and give
necessary and sufficient conditions for a solution under the additional
hypothesis that $\phi'(0)$ is diagonalizable.  This talk will use basic
concepts of linear algebra, complex analysis, and compact operators to give
necessary and sufficient conditions for a Schroeder solution under the
general hypotheses.