Maps strongly preserving the square zero of $\lambda$-Lie product of operators

Document Type : Original Article

Author

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P. O. Box 47416-1468, Babolsar, Iran.

10.22034/kjm.2020.210055.1640

Abstract

Let $\mathcal{A}$ be a standard operator algebra on a Banach space $\mathcal{X}$ with $\dim \mathcal{X}\geq 2$. In this paper, we characterize
the forms of additive maps on $\mathcal{A}$ which strongly preserve the
square zero of $\lambda$-Lie product of operators, i.e., if $\phi:\mathcal{A}\longrightarrow \mathcal{A}$ is an additive map which satisfies
$$[A,B]^2_{\lambda}=0 \Rightarrow [\phi(A),B]^2_{\lambda}=0,$$
for every $A,B \in \mathcal{A}$ and for a scalar number $\lambda$ with $\lambda \neq -1$, then it is shown that
there exists a function $\sigma: \mathcal{A} \rightarrow \mathbb{C}$ such that $\phi(A)= \sigma(A) A$, for every $A \in \mathcal{A}$.

Keywords