New insights into linear maps which are anti-derivable at zero

Let $A$ be a Banach algebra admitting a bounded approximate unit and satisfying property $\mathbb{B}$. Suppose $T: A \rightarrow X$ is a continuous linear map, where $X$ is an essential Banach $A$-bimodule. We prove that the following statements are equivalent: $(i)$ $T$ is anti-derivable at zero (i.e., $a b =0$ in $A$ $\Rightarrow T(b)\cdot a + b\cdot T(a) =0$); $(ii)$ There exist an element $\xi \in X^{**}$ and a linear map (actually a bounded Jordan derivation) $d: A\to X$ satisfying $\xi \cdot a = a \cdot \xi \in X$, $T(a) = d(a) +\xi \cdot a$, and $d(b)\cdot a + b\cdot d(a)= - 2 \xi \cdot (b a),$ for all $a,b\in A$ with $a b =0$. Assuming that $A$ is a C$^*$-algebra we show that a bounded linear mapping $T: A\to X$ is anti-derivable at zero if, and only if, there exist an element $\eta \in X^{**}$ and an anti-derivation $d: A \rightarrow X$ satisfying $\eta \cdot a = a \cdot \eta \in X$, $\eta \cdot [a,b] = 0$ {\rm(}i.e., $L_{\eta}: A \to A$, $L_{\eta} (a) = \eta \cdot a$ vanishes on commutators{\rm)}, and $T(a) = d(a) +\eta \cdot a$, for all $a,b \in A$. The results are also applied for some special operator algebras.