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  Volume 6, Issue 2, Article 49
Characterization of the Trace by Young's Inequality

    Authors: A.M. Bikchentaev, O.E. Tikhonov,  
    Keywords: Characterization of the trace, Matrix Young's inequality.  
    Date Received: 25/03/05  
    Date Accepted: 11/04/05  
    Subject Codes:


    Editors: Tsuyoshi Ando,  

Let $ varphi$ be a positive linear functional on the algebra of $ n times n$ complex matrices and $ p$, $ q$ be positive numbers such that $ frac{1}{p}+frac{1}{q}=1$. We prove that if for any pair $ A$, $ B$ of positive semi-definite $ n times n$ matrices the inequality

$displaystyle varphi (vert ABvert) le frac{varphi (A^p)}{p} + frac{varphi (B^q)}{q} $
holds, then $ varphi$ is a positive scalar multiple of the trace.

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