Order relations for the near geometric and near spectral geometric curves
We investigate order and spectral properties of the weighted near geometric curve $A\star_tB=A^{1/2}(B^{1/2}A^{-1}B^{1/2})^tA^{1/2}$ 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 $A\widetilde{\natural}_tB=(A^{-1}\star B)^tA(A^{-1}\star B)^t$, where $\star=\star_{1/2}$. Setting $D=A^{-1}\star_1B$, we prove that $A\widetilde{\natural}_tB=A\natural_tD$ and $B\prec_{\log}D$, 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.