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.

中文翻译：

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保留了$$ B（{\ mathcal {H}}）$$ B（H）上标量的r次幂微扰的加性映射

设\（{\ mathcal {H}} \），\（{\ mathcal {K}} \）是\（{\ mathbb {F}} \）上\（\ dim {\ mathcal {H} } \ ge 3 \），其中\（{\ mathbb {F}} \）是实数字段或复数字段。假设\（\ varphi：{B}（{\ mathcal {H}}）\ rightarrow {B}（{\ mathcal {K}}）\）是加性射影图，而\（r \ ge 3 \）是一个正整数。结果表明，\（\ varphi \）是r-在两个方向上都保留的标量的幂幂扰动，当且仅当\（\ varphi（A）= cTAT ^ {-1} + g（A）I \）满足每个\（A \ in {B}（{\ mathcal {H}}）\）；要么\（\ varphi（A）= cTA ^ {*} T ^ {-1} + g（A）I \）对于每个\（A \ in {B}（{\ mathcal {H}}）\）成立，其中\ {0 \ not = c \ in {{\ mathbb {F}}} \）中，\（T：{\ mathcal {H}} \ rightarrow {\ mathcal {K}} \）是\（\ tau \）-具有\（\ tau：{\ mathbb {F}} \ rightarrow {\ mathbb {F}} \）的线性双射映象，并且g是\（B（{\ mathcal {H}} ）\）放入\（{{\ mathbb {F}}} \\）中。作为应用，对于任何整数\（k \ ge 5 \），加法运算\（{B}（{\ mathcal {H}}）\）上的加k交换交换图和一般完全k交换交换图。 分别进行表征。