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## Characterization of Jordan derivations on $\mathcal J$-subspace lattice algebras

### Tom 210 / 2012

Studia Mathematica 210 (2012), 17-33 MSC: Primary 47L35; Secondary 47B47. DOI: 10.4064/sm210-1-2

#### Streszczenie

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.

#### Autorzy

• Xiaofei QiDepartment of Mathematics
Shanxi University
030006 Taiyuan, P.R. China
e-mail

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