Putnam-Fuglede交换性与初等算子的值域-核正交性
Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator
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中文总结 AI 辅助
本文针对希尔伯特空间上的初等算子，证明了其核的包含关系与值域-核正交性的充要条件，并将结果推广到一类含可交换系数的初等算子，给出了对应的正交性结论。
中文摘要 AI 辅助
设希尔伯特空间H上的可交换算子T、S属于ℬ(H)，其中T是w-亚正规算子且满足ker T ⊆ ker T^*，S是正规算子。令φ_{T,S} ∈ ℬ(ℬ(H))为由φ_{T,S}(X)=T X S^*-S X T^*定义的初等算子。本文首先证明：(1) ker φ_{T,S} ⊂ ker φ_{T^*,S^*}；(2) φ_{T,S}的值域与φ_{T,S}的核正交（ℛ(φ_{T,S}) ⊥ ker φ_{T,S}）当且仅当ker T ∩ ker S={0}。其次，将这些结果推广到由Φ(X)=A X D-C X B定义的初等算子Φ ∈ ℬ(ℬ(H))，其中[A,C]=[B,D]=0，同时给出了初等算子Φ的相关正交性结果。
英文摘要
Given Hilbert space commuting operators $T, S \in \mcl(H)$, such that $T$ is $w$-hyponormal with $\ker T \subseteq \ker T^*$ and $S$ is normal operator. Let $ϕ_{T, S} \in \mcl(\mcl(H))$ be the elementary operator defined by $ϕ_{T, S} (X) = T X S^*-S X T^*$. In this paper, we show firstly that (1) $\ker ϕ_{T, S} \subset \ker ϕ_{T^*, S^*}$. (2) The range of $ϕ_{T, S}$ is orthogonal to the kernel of $ϕ_{T, S}$ ( $ \mcr(ϕ_{T, S}) \perp \ker ϕ_{T, S} $ ) if and only if $\ker T \cap \ker S=\{0\}$. Secondly, we will extend these results to the elementary operator $Φ\in \mcl(\mcl(H))$ defined by $\;Φ(X)=A X D-C X B$ where $[A, C]=[B, D]= 0$. Related orthogonality results for the elementary operator $Φ$ are also given.
