Nonlinear maps preserving certain subspaces of Lie product of operators

Abstract

In this paper, we describe all surjective maps $\phi$ on ${\mathcal {B}}(X)$, the space of all bounded linear operators on an infinite-dimensional complex Banach space X, which satisfy

$$\begin{aligned} H_{0}(\phi (T)\phi (A) - \phi (A)\phi (T))= H_{0}(TA-AT) \end{aligned}$$

for all $A,T \in {\mathcal {B}}(X).$ Similarly for surjective maps $\phi$ on ${\mathcal {B}}(X)$ satisfying

$$\begin{aligned} K(\phi (T)\phi (A) - \phi (A)\phi (T))= K(TA-AT) \end{aligned}$$

for all $A,T \in {\mathcal {B}}(X).$