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Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator
Abstract
Given Hilbert space commuting operators T, S ∈ mcl(H), such that T is w-hyponormal with ker T ⊆ ker T* and S is normal operator. Let ϕT, S ∈ mcl(mcl(H)) be the elementary operator defined by ϕT, S (X) = T X S*−S X T*. In this paper, we show firstly that (1) ker ϕT, S ⊂ ker ϕT*, S*. (2) The range of ϕT, S is orthogonal to the kernel of ϕT, S ( $ \mcr(ϕ_{T, S}) \perp \ker ϕ_{T, S} $ ) if and only if ker T ∩ ker S={0}. Secondly, we will extend these results to the elementary operator Φ∈ 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.
Related
- Orthograph related to mutual (strong) numerical radius orthogonality on operator algebra
- Hahn-Type Orthogonality for the Resolvent Delta Operator ℒ=D(I−D)−1
- Elementary cellular automata: spectral properties of Koopman operator for finite periodic realizations
- Fuglede's theorem in the Heisenberg group
- Fuglede's Conjecture for Cyclic Groups of Square-free Order
BibTeX
@misc{morjane2026putnamfuglede,
title = {{Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator}},
author = {M. Morjane and M. Ech-chad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18125},
eprint = {2608.18125},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18125},
}
RIS
TY - UNPB
TI - Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator
AU - M. Morjane
AU - M. Ech-chad
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2608.18125
UR - https://arxiv.org/abs/2608.18125
AB - Given Hilbert space commuting operators T, S ∈ mcl(H), such that T is w-hyponormal with ker T ⊆ ker T^* and S is normal operator. Let ϕ_T, S ∈ 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 ⊂ 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 ∩ ker S={0}. Secondly, we will extend these results to the elementary operator Φ∈ 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.
ER -