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Abstract

Let $ \mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a Banach space $X$ and $\mathop{\rm Alg}\nolimits \mathcal{L}$ the associated $\mathcal{J}$-subspace lattice algebra. Assume that $\delta:\mathop{\rm Alg}\nolimits \mathcal{L}\rightarrow\mathop{\rm Alg}\nolimits \mathcal{L}$ is an additive map. It is shown that $\delta$ satisfies $\delta(AB+BA)=\delta(A)B+A\delta(B)+\delta(B)A+B\delta(A)$ for any $A,B\in\mathop{\rm Alg}\nolimits \mathcal{L}$ with $AB+BA=0$ if and only if $\delta(A)=\tau(A)+\delta(I)A$ for all $A$, where $\tau$ is an additive derivation; if $X$ is complex with $\dim X\geq 3$ and if $\delta$ is linear, then $\delta$ satisfies $\delta(AB+BA)=\delta(A)B+A\delta(B)+\delta(B)A+B\delta(A)$ for any $A,B\in\mathop{\rm Alg}\nolimits \mathcal{L}$ with $AB+BA=I$ if and only if $\delta$ is a derivation.