The commutant of an analytic Toeplitz operator

Abstract

For a function f in H ∞ {H^\infty } of the unit disk, the operator on H 2 {H^2} of multiplication by f will be denoted by T f {T_f} and its commutant by { T f } ′ \{ {T_f}\} ’ . For a finite Blaschke product B, a representation of an operator in { T B } ′ {\{ {T_B}\}’} as a function on the Riemann surface of B − 1 ∘ B {B^{ - 1}} \circ B motivates work on more general functions. A theorem is proved which gives conditions on a family F \mathcal {F} of H ∞ {H^\infty } functions which imply that there is a function h such that { T h } ′ = ∩ f ∈ F { T f } ′ \{ {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 ) 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 } ′ \{ {T_f}\} ’ = \{ {T_B}\} ’ . Necessary and sufficient conditions are given for an operator to commute with T f {T_f} when f is a covering map (in the sense of Riemann surfaces). If f and g are in H ∞ {H^\infty } and f = h ∘ g f = h \circ g , then { T f } ′ ⊃ { T g } ′ \{ {T_f}\} ’ \supset \{ {T_g}\} ’ . This paper introduces a class of functions, the H 2 {H^2} -ancestral functions, for which the converse is true. If f and g are H 2 {H^2} -ancestral functions, then { T f } ′ ≠ { T g } ′ \{ {T_f}\} ’ \ne \{ {T_g}\} ’ unless f = h ∘ g f = h \circ g where h is univalent. It is shown that inner functions and covering maps are H 2 {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 {T_f} does not commute with nonzero compact operators. It follows from one of these results that if f is an H 2 {H^2} -ancestral function, then T f {T_f} does not commute with any nonzero compact operators.