Intersections of commutants of analytic Toeplitz operators

In this paper we study the intersection of commutants of analytic Toeplitz operators. Our main result is that if $\phi$ is a finite Blaschke product and $\Psi \epsilon {H^\infty }$, then $\{ {T_\phi }\} ' \cap \{ {T_\Psi }\} ' = \{ {T_I}\} '$ where $I$ is a finite Blaschke product and $\phi$ and $\Psi$ are functions of $I$. The key step is a function-theoretic theorem describing the relationship between a finite Blaschke product and any ${H^\infty }$ function.