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  Volume 6, Issue 3, Article 87
A Minkowski-Type Inequality for the Schatten Norm

    Authors: Markus Sigg,  
    Keywords: Schatten class, Schatten norm, Norm inequality, Minkowski inequality, Triangle inequality, Powers of operators, Schatten-Minkowski constant.  
    Date Received: 16/07/04  
    Date Accepted: 29/06/05  
    Subject Codes:

47A30, 47B10

    Editors: Frank Hansen,  

Let $ F$ be a Schatten $ p$-operator and $ R,S$ positive operators. We show that the inequality $ vert{F (R+S)^frac1cvert _p}^{hspace{-4.75pt}c} le { vert FR^frac1cvert _p}^{hspace{-4.75pt}c} + {vert FS^frac1cvert _p}^{hspace{-4.75pt}c}$ for the Schatten $ p$-norm $ vertcdot vert _p$ is true for $ p ge c = 1$ and for $ p ge c = 2$, conjecture it to be true for $ p ge c in [1,2]$, give counterexamples for the other cases, and present a numerical study for $ 2 times 2$ matrices. Furthermore, we have a look at a generalisation of the inequality which involves an additional factor $ sigma(c,p)$.

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