Contractive operators
T
T
that are trace class perturbations of a unitary operator
U
U
are treated. It is proved that the dimension functions of the absolutely continuous spectrum of
T
T
,
T
∗
T^*
, and of
U
U
coincide. In particular, if
U
U
has a purely singular spectrum, then the characteristic function
θ
\theta
of
T
T
is a two-sided inner function, i.e.,
θ
(
ξ
)
\theta (\xi )
is unitary a.e. on
T
\mathbb {T}
. Some corollaries to this result are related to investigations of the asymptotic stability of the operators
T
T
and
T
∗
T^*
(the convergence
T
n
→
0
T^n\to 0
and
(
T
∗
)
n
→
0
(T^*)^n\to 0
, respectively, in the strong operator topology).
The proof is based on an explicit computation of the characteristic function.