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

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.