Let
G
G
be a locally compact abelian group and
X
X
be a Banach space. Let
L
1
(
G
,
X
)
{L^1}(G,X)
be the Banach space of
X
X
-valued Bochner integrable functions on
G
G
. We prove that the space of bounded linear translation invariant operators of
L
1
(
G
,
X
)
{L^1}(G,X)
can be identified with
L
(
X
,
M
(
G
,
X
)
)
L(X,M(G,X))
, the space of bounded linear operators from
X
X
into
M
(
G
,
X
)
M(G,X)
where
M
(
G
,
X
)
M(G,X)
is the space of
X
X
-valued regular, Borel measures of bounded variation on
G
G
. Furthermore, if
A
A
is a commutative semisimple Banach algebra with identity of norm 1 then
L
1
(
G
,
A
)
{L^1}(G,A)
is a Banach algebra and we prove that the space of multipliers of
L
1
(
G
,
A
)
{L^1}(G,A)
is isometrically isomorphic to
M
(
G
,
A
)
M(G,A)
. It also follows that if dimension of
A
A
is greater than one then there exist translationinvariant operators of
L
1
(
G
,
A
)
{L^1}(G,A)
which are not multipliers of
L
1
(
G
,
A
)
{L^1}(G,A)
.