On two geometric means and sum of adjoint orbits

Gan, Luyining; Liu, Xuhua; Tam, Tin-Yau

doi:10.1016/j.laa.2021.08.029

Cited by 6 publications

(6 citation statements)

References 29 publications

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“…Generalizations to semisimple Lie groups. The readers are referred to [8,9] regarding Lie theoretic preliminaries and notations. Let G be a noncompact connected semisimple Lie group with Lie algebra g. Let Θ : G → G be a Cartan involution of G, and let K be the fixed point set of Θ, which is is an analytic subgroup of G. Denote by θ the differential map dΘ of Θ.…”

Section: According To the Riccati Equation It Implies Thatmentioning

confidence: 99%

“…They also studied several properties similar to those of the Tsallis relative operator entropy by the usual geometric mean. Recently, Gan, Liu, and Tam [8] and Gan and Tam [9] studied A t B and obtained some nice properties. A nice survey and new properties related to the weighted spectral geometric mean A t B can be found in the recent paper of Gan and Kim [7].…”

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confidence: 99%

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A new weighted spectral geometric mean and properties

Dinh,

Tam,

Vuong

2024

ELA

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In this paper, we introduce a new weighted spectral geometric mean: \begin{equation*}\label{F-mean}F_t(A,B)= (A^{-1}\sharp_t B)^{1/2} A^{2-2t} (A^{-1} \sharp_t B)^{1/2}, \quad t\in [0,1],\end{equation*} where $A$ and $B$ are positive definite matrices. We study basic properties and inequalities for $F_t(A, B)$. We also establish the Lie-Trotter formula for $F_t(A, B)$. Finally, we extend some of the results on $F_t(A, B)$ to symmetric space of noncompact types.

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Section: According To the Riccati Equation It Implies Thatmentioning

confidence: 99%

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confidence: 99%

A new weighted spectral geometric mean and properties

Dinh,

Tam,

Vuong

2024

ELA

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“…We say that A is positively similar to B if there exists C ∈ P n such that A = CBC −1 . Let us first recall some basic properties of the t-spectral mean [7,10,22] in the following theorem.…”

Section: Propertiesmentioning

confidence: 99%

“…Very recently Gan, Liu and Tam [10] have proved the following result which asserts that the t-geometric mean of two positive definite matrices is log majorized by their t-spectral mean.…”

Section: Log Majorizationmentioning

confidence: 99%

“…It is the unique geodesic in P from p (at t = 0) to q (at t = 1). When t = 1/2, we abbreviate p 1/2 q as p q, Similarly, the t-spectral mean of p, q ∈ P was defined by Gan, Liu and Tam [10]: p t q = (p −1 q) t p(p −1 q) t , 0 t 1. One may interpret p t q as the outcome of the action of (p −1 q) t on p in view of the Gaction (5.5) on P .…”

Section: Kostant's Pre-order and Symmetric Spacesmentioning

confidence: 99%

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Inequalities and limits of weighted spectral geometric mean

Gan¹,

Tam²

2021

Preprint

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We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between B ts/2 A (1−t)s B ts/2 1/s and the t-spectral mean A t B := (A −1 B) t A(A −1 B) t of two positive semidefinite matrices A and B, where A B is the geometric mean, and the t-spectral mean is the dominant one. The limit involving t-spectral mean is also studied. We then extend all the results in the context of symmetric spaces of negative curvature.

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