Order relations for the near geometric and near spectral geometric curves

@misc{ibrahim2026order,
  title = {{Order relations for the near geometric and near spectral geometric curves}},
  author = {Muritala Abiodun Ibrahim and Huajun Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23736},
  eprint = {2610.23736},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23736},
}

TY  - UNPB
TI  - Order relations for the near geometric and near spectral geometric curves
AU  - Muritala Abiodun Ibrahim
AU  - Huajun Huang
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23736
UR  - https://arxiv.org/abs/2610.23736
AB  - We investigate order and spectral properties of the weighted near geometric curve A⋆_tB=A^1/2(B^1/2A^−1B^1/2)^tA^1/2 on the cone of positive definite matrices. Our focus is on comparisons governed by the Loewner order, eigenvalue inequalities, and log-majorization. We establish a normalized eigenvalue comparison with the weighted arithmetic mean, give a sufficient condition for the corresponding Loewner inequality, and obtain conditional log-majorization relations. Motivated by the relationship between the weighted geometric and spectral geometric means, we introduce the near spectral geometric curve Anatural̃_tB=(A^−1⋆ B)^tA(A^−1⋆ B)^t, where ⋆=⋆_1/2. Setting D=A^−1⋆_1B, we prove that Anatural̃_tB=Anatural_tD and B≺_logD, and show that D=B if and only if A and B commute. We further characterize commutativity and near-order monotonicity, prove midpoint isospectrality, and derive affine interpolation and constant-speed identities. Finally, we establish a Lie-Trotter formula and additional log-majorization inequalities. These results connect the two near geometric constructions with the geometric and spectral geometric means and clarify their order and spectral behavior.
ER  -