A REVERSE INEQUALITY FOR SOME GENERALIZED GEOMETRIC MEANS INVOLVING UNITAL POSITIVE LINEAR MAPS

Subject Areas : Statistics

Fatemeh Khosravi ¹ ( Department of Mathematics, Lorestan University, P.O.Box 465, Khoramabad, Iran. )
Amir ghasem Ghazanfari ² ( Department of Mathematics Lorestan University P. O. Box 465, Khoramabad, Iran. )

Received: 2020-07-25 Accepted : 2020-09-28 Published : 2022-01-21

Keywords: نگاشتهای خطی مثبت, میانگین های هندسی, شعاع طیفی, میانگین های توانی,

Abstract :

Let $B(H)$ be the $C^*$-algebra of all bounded linear operators on a complex Hilbert spaces $H$. Ando, Li and Mathias introduced a generalized geometric mean for $n$ positive definite operators. This geometric mean $G(A_{1},A_{2},dots ,A_{n})$ of any $n$-tuple of positive definite operators $mathbb{A}=(A_1,dots,A_n)$ is defined by induction.(i) $G(A_{1},A_{2})$ =$A_{1}sharp A_{2}$ (ii) Assume that the geometric mean any $(n-1)$-tuple of operators is defined. Let [G ((A_j)_{jneq i })= G(A_{1},A_{2},dots,A_{i-1},A_{i+1},dots,A_{n}), ] and let sequences ${mathbb{A}_{i}^{(r)}} _{r=1}^{infty}$ be $mathbb{A}_{i}^{(1)}= A_{i}$ and $ mathbb{A}_{i}^{(r+1)}=G((mathbb{A}_{j}^{(r)})_{jneq i }) $. If there exists $ lim_{rrightarrowinfty}{mathbb{A}_{i}^{(r)}} $, and it does not depend on $i$. Hence the geometric mean of $n$-operators is defined by begin{equation*} lim_{rrightarrowinfty}{mathbb{A}_{i}^{(r)}} = G((mathbb{A}))=G(A_{1},A_{2},dots,A_{n}) text{ for } i= 1,dots,n. end{equation*} We shall show a reverse inequality for the generalized geometric mean defined by Ando-Li-Mathias for $n$ positive definite operators, as follows: Let $Phi$ be a unital positive linear map on $B(H)$ and $r(A)$ be the spectral radius of $A$, then begin{align*} Phi(G(A_1,dots,A_n))geqleft(frac{2h}{1+h^2}right)^{n-1}G(Phi(A_1),dots,Phi(A_n)), end{align*} where $R(A_i, A_j)=max{r(A_i^{-1}A_j) ,r(A_j^{-1}A_i)}$ and $h=min_{i,j} R(A_i, A_j)$.The Karcher mean, also called the Riemannian mean, Recently it has been used in a diverse variety of settings:diffusion tensors in medical imaging and radar, covariance matrices in statistics, kernels in machine learning andelasticity. We also give a reverse inequality for the weighted power mean of $n$ positive definite operators involving unital positive linear maps.

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