The commutant of analytic Toeplitz operators

In this paper we study the commutant of an analytic Toeplitz operator. For $\phi \;\;{H^\infty }$, let $\phi = \chi F$ be its inner-outer factorization. Our main result is that if there exists $\lambda \;\epsilon \;{\text{C}}$ such that X factors as $\chi = {\chi _1}{\chi _2} \cdots {\chi _n}$, each ${\chi _i}$ an inner function, and if $F - \lambda$ is divisible by each ${\chi _i}$, then $\{ {T_\phi }\} ' = \{ {T_\chi }\} ' \cap \{ {T_F}\} '$. The key step in the proof is Lemma 2, which is a curious result about nilpotent operators. One corollary of our main result is that if $\chi (z) = {z^n},n \geq 1$, then $\{ {T_\phi }\} ' = \{ {T_\chi }\} ' \cap \{ {T_F}\} '$, another is that if $\phi \;\epsilon {H^\infty }$ is univalent then $\{ {T_\phi }\} ' = \{ {T_z}\} '$. We are also able to prove that if the inner factor of $\phi$ is $\chi (z) = {z^n},n \geq 1$, then $\{ {T_\phi }\} ' = \{ {T_{{z^s}}}\} '$ where s is a positive integer maximal with respect to the property that ${z^n}$ and $F(z)$ are both functions of ${z^s}$. We conclude by raising six questions.