Linear maps which are anti-derivable at zero

Let $T:A\to X$ be a bounded linear operator, where $A$ is a C${}^{\ast }$-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. $ab=0$ in $A$ implies $T(b) a + b T(a)=0$);$(b)$Thereexistananti-derivation$d:A\to X^{}$andanelement$\xi \in X^{}$satisfying$\xi a = a \xi,$ $\xi [a,b]=0,$ $T(a b) = b T(a) + T(b) a - b \xi a,$and$T(a) = d(a) + \xi a,$forall$a,b\in A$. We also prove a similar equivalence when $X$ is replaced with $A^{\ast \ast }$. This provides a complete characterization of those bounded linear maps from $A$ into $X$ or into $A^{\ast \ast }$ which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are ${}^{\ast }$-anti-derivable at zero.