Additive maps preserving r-nilpotent perturbation of scalars on $B({\mathcal {H}})$

Abstract

Let ${\mathcal {H}}$, ${\mathcal {K}}$ be Hilbert spaces over ${\mathbb {F}}$ with $\dim {\mathcal {H}}\ge 3$, where ${\mathbb {F}}$ is the real or complex field. Assume that $\varphi :{B}({\mathcal {H}})\rightarrow {B}({\mathcal {K}})$ is an additive surjective map and $r\ge 3$ is a positive integer. It is shown that $\varphi$ is r-nilpotent perturbation of scalars preserving in both directions if and only if either $\varphi (A)=cTAT^{-1}+g(A)I$ holds for every $A\in {B}({\mathcal {H}})$; or $\varphi (A)=cTA^{*}T^{-1}+g(A)I$ holds for every $A\in {B}({\mathcal {H}})$, where $0\not =c\in {{\mathbb {F}}}$, $T:{\mathcal {H}}\rightarrow {\mathcal {K}}$ is a $\tau$-linear bijective map with $\tau :{\mathbb {F}}\rightarrow {\mathbb {F}}$ an automorphism and g is an additive map from $B({\mathcal {H}})$ into ${{\mathbb {F}}}$. As applications, for any integer $k\ge 5$, additive k-commutativity preserving maps and general completely k-commutativity preserving maps on ${B}({\mathcal {H}})$ are characterized, respectively.