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  Volume 6, Issue 5, Article 139
 
Hermitian Operators and Convex Functions
    Authors: Jean-Christophe Bourin,  
    Keywords: Hermitian operators, eigenvalues, operator inequalities, Jensen's inequality.  
    Date Received: 06/04/05  
    Date Accepted: 10/11/05  
    Subject Codes:

47A30 47A63.

  
    Editors: Frank Hansen,  
 
    Abstract:

We establish several convexity results for Hermitian matrices. For instance: Let $ A$, $ B$ be Hermitian and let $ f$ be a convex function. If $ X$ and $ Y$ stand for $ f({A+B}/2)$ and $ {f(A)+f(B)}/2$ respectively, then there exist unitaries $ U$, $ V$ such that

$displaystyle Xle frac{UYU^* + VYV^*}{2}.$    
Consequently, $ lambda_{2j-1}(X) le lambda_j(Y)$, where $ lambda_j(cdot)$ are the eigenvalues arranged in decreasing order.

         
       
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