Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Morjane, M., Ech-chad, M.
Institution: Université Ibn-Tofail
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203919348