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.