论文标题
保留操作员偏斜产品的局部光谱子空间的地图
Maps preserving the local spectral subspace of skew-product of operators
论文作者
论文摘要
令$ b（h）$为无限尺寸复杂的希尔伯特·太空$ h $上的所有有界线性操作员的代数。对于$ t \ in b（h）$和$λ\ in \ mathbb {c} $，令$ h_ {t}（\ {λ\}）$表示与$ \ {λ\ \} $相关的$ t $的本地频谱子空间。我们证明，如果$φ：b（h）\ rightarrow b（h）$是一个加性地图，以使其范围最多包含所有排名最多的运算符，并且满足$ h_ {φ（t）φ（s）^{\ ast}}}}}}}}}}（\ {λ\\}） $ t，s \ in b（h）$和$λ\在\ mathbb {c} $中，然后在$ b（h）$中存在一个单一操作员$ v $和一个非零的标量$μ$，使得$φ（t）=μtv^{\ ast} $ in B（h）$ in b（h）$。我们还显示，如果$φ_{1} $和$φ_{2} $是$ b（h）$的加性映射到$ b（h）$中，以使他们的范围最多包含所有排名的所有操作员，并且满足$ h_ {φ_{φ_{1}（1}（t）（t）（t）φ_{2}}（s） h_ {ts^{\ ast}}（\ {λ\}）$$用于b（h）$中的所有$ t，s \ in \ mathbb {c} $ in b（h）$中。然后$φ_{2}（i）^{\ ast} $是可逆的，$φ_{1}（t）= t（φ_{2}（i）^{\ ast}）^{\ ast}）^{ - 1} $和$φ____________________{2}（t）= m $ { b（h）$。
Let $B(H)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $H$. For $T \in B(H)$ and $λ\in \mathbb{C}$, let $H_{T}(\{λ\})$ denotes the local spectral subspace of $T$ associated with $\{λ\}$. We prove that if $φ:B(H)\rightarrow B(H)$ be an additive map such that its range contains all operators of rank at most two and satisfies $$H_{φ(T)φ(S)^{\ast}}(\{λ\})= H_{TS^{\ast}}(\{λ\})$$ for all $T, S \in B(H)$ and $λ\in \mathbb{C}$, then there exist a unitary operator $V$ in $B(H)$ and a nonzero scalar $μ$ such that $φ(T) = μTV^{\ast}$ for all $T \in B(H)$. We also show if $φ_{1}$ and $φ_{2}$ be additive maps from $B(H)$ into $B(H)$ such that their ranges contain all operators of rank at most two and satisfies $$H_{φ_{1}(T)φ_{2}(S)^{\ast}}(\{λ\})= H_{TS^{\ast}}(\{λ\})$$ for all $T, S \in B(H)$ and $λ\in \mathbb{C}$. Then $φ_{2}(I)^{\ast}$ is invertible, and $φ_{1}(T) = T(φ_{2}(I)^{\ast})^{-1}$ and $φ_{2}(T) =φ_{2}(I)^{\ast}T$ for all $T \in B(H)$.
