An operator
T
u
g
{T_{ug}}
on a Hilbert space
H
H
of functions on a set
X
X
defined by
T
u
g
(
f
)
=
u
(
f
∘
g
)
{T_{ug}}(f) = u(f \circ g)
, where
f
f
is in
H
,
u
:
X
→
C
H,\;u:X \to {\mathbf {C}}
and
g
:
X
→
X
g:X \to X
, is called a weighted composition operator. In this paper
X
X
is the set of integers and
H
=
L
2
(
Z
,
μ
)
H = {L^2}({\mathbf {Z}},\mu )
, where
μ
\mu
is a measure whose sigma-algebra is the power set of
Z
{\mathbf {Z}}
. One distinguished space is
l
2
=
L
2
(
Z
,
μ
)
{l^2} = {L^2}({\mathbf {Z}},\mu )
, where
μ
\mu
is counting measure. The most important results given here are the determination of the spectrum of
T
u
g
{T_{ug}}
on
l
2
{l^2}
and a characterization of the commutant of
T
g
{T_g}
on
L
2
(
Z
,
μ
)
{L^2}({\mathbf {Z}},\mu )
. To obtain many of the results it was necessary to assume the function
g
g
to be one-to-one except on a finite subset of the integers.