We give an example of a subnormal operator T such that
C
∖
σ
(
T
)
{\text {C}}\,\backslash \,\sigma (T)
has an infinite number of components,
int
(
σ
(
T
)
)
\operatorname {int} (\sigma (T))
has two components U and V, and T cannot be decomposed with respect to U and V. That is, it is impossible to write
T
=
T
1
⊕
T
2
T\, = \,{T_1}\, \oplus \,{T_2}
with
σ
(
T
1
)
=
U
¯
\sigma ({T_1})\, = \,\overline U
and
σ
(
T
2
)
=
V
¯
\sigma ({T_2})\, = \,\overline V
. This example shows that Sarason’s decomposition theorem cannot be extended to the infinitely-connected case. We also use Mlak’s generalization of Sarason’s theorem to prove theorems on the existence of reducing subspaces. For example, if X is a spectral set for T and
K
⊂
X
K\, \subset \,X
, conditions are given which imply that T has a nontrivial reducing subspace
M
\mathcal {M}
such that
σ
(
T
|
M
)
⊂
K
\sigma (T|\mathcal {M})\, \subset \,K
. In particular, we show that if T is a subnormal operator and if
Γ
\Gamma
is a piecewise
C
2
{C^2}
Jordan closed curve which intersects
σ
(
T
)
\sigma (T)
in a set of measure zero on
Γ
\Gamma
, then
T
=
T
1
⊕
T
2
T\, = \,{T_1}\, \oplus \,{T_2}
with
σ
(
T
1
)
⊂
σ
(
T
)
∩
ext
(
Γ
)
¯
\sigma ({T_1})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname {ext} (\Gamma )}
and
σ
(
T
2
)
⊂
σ
(
T
)
∩
int
(
Γ
)
¯
\sigma ({T_2})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname {int} (\Gamma )}
.