Abstract
Let $$T:A\rightarrow X$$
T
:
A
→
X
be a bounded linear operator, where A is a $$\hbox {C}^*$$
C
∗
-algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent:
(a)
T is anti-derivable at zero (i.e., $$ab =0$$
a
b
=
0
in A implies $$T(b) a + b T(a)=0$$
T
(
b
)
a
+
b
T
(
a
)
=
0
);
(b)
There exist an anti-derivation $$d:A\rightarrow X^{**}$$
d
:
A
→
X
∗
∗
and an element $$\xi \in X^{**}$$
ξ
∈
X
∗
∗
satisfying $$\xi a = a \xi ,$$
ξ
a
=
a
ξ
,
$$\xi [a,b]=0,$$
ξ
[
a
,
b
]
=
0
,
$$T(a b) = b T(a) + T(b) a - b \xi a,$$
T
(
a
b
)
=
b
T
(
a
)
+
T
(
b
)
a
-
b
ξ
a
,
and $$T(a) = d(a) + \xi a,$$
T
(
a
)
=
d
(
a
)
+
ξ
a
,
for all $$a,b\in A$$
a
,
b
∈
A
.
We also prove a similar equivalence when X is replaced with $$A^{**}$$
A
∗
∗
. This provides a complete characterization of those bounded linear maps from A into X or into $$A^{**}$$
A
∗
∗
which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are $$^*$$
∗
-anti-derivable at zero.
On the surjectivity of linear maps on locally convex spaces
