The commutant of an analytic Toeplitz operator

For a function f in ${H^\infty }$ of the unit disk, the operator on ${H^2}$ of multiplication by f will be denoted by ${T_f}$ and its commutant by $\{ {T_f}\} ’$. For a finite Blaschke product B, a representation of an operator in ${\{ {T_B}\}’}$ as a function on the Riemann surface of ${B^{ - 1}} \circ B$ motivates work on more general functions. A theorem is proved which gives conditions on a family $\mathcal {F}$ of ${H^\infty }$ functions which imply that there is a function h such that $\{ {T_h}\} ’ = { \cap _{f \in \mathcal {F}}}\{ {T_f}\} ’$. As a special case of this theorem, we find that if the inner factor of $f - f(c)$ is a finite Blaschke product for some c in the disk, then there is a finite Blaschke product B with $\{ {T_f}\} ’ = \{ {T_B}\} ’$. Necessary and sufficient conditions are given for an operator to commute with ${T_f}$ when f is a covering map (in the sense of Riemann surfaces). If f and g are in ${H^\infty }$ and $f = h \circ g$, then $\{ {T_f}\} ’ \supset \{ {T_g}\} ’$. This paper introduces a class of functions, the ${H^2}$-ancestral functions, for which the converse is true. If f and g are ${H^2}$-ancestral functions, then $\{ {T_f}\} ’ \ne \{ {T_g}\} ’$ unless $f = h \circ g$ where h is univalent. It is shown that inner functions and covering maps are ${H^2}$-ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function f which imply that ${T_f}$ does not commute with nonzero compact operators. It follows from one of these results that if f is an ${H^2}$-ancestral function, then ${T_f}$ does not commute with any nonzero compact operators.