JIPAM

On an Inequality Involving Power and Contraction of Matrices with and without Trace  
 
  Authors: Marcos V. Travaglia,  
  Keywords: Trace inequalities, Operator inequalities, Positive semidefinite matrix, Operator monotony, Operator concavity.  
  Date Received: 28/01/06  
  Date Accepted: 16/02/06  
  Subject Codes:

15A45, 15A90, 47A63.

  
  Editors: Frank Hansen,  
 
  Abstract:

Let $A$ and $B$ be positive semidefinite matrices. Assuming that the eigenvalues of $B$ are less than one, we prove the following trace inequalities

begin{displaymath} mathrm{Tr}left{ (BA^{alpha }B)^{1/alpha} right} leq mathrm{Tr}left{ (BA^{beta }B)^{1/beta} right} end{displaymath}

and
begin{displaymath} mathrm{Tr}left{ left( BA^{alpha }B right)^{1/alpha} ... ...eta} A^{alpha} B^{alpha/beta} right)^{1/alpha} right} , end{displaymath}

for all begin{displaymath} ( B A^alpha B )^{1/ alpha}  leq  ( B A^beta B )^{1/beta} end{displaymath}
in the cases (a) $1 leq alpha leq beta$ or (b) $frac{1}{2} leq alpha leq beta$ and $beta geq 1$. Further we present counterexamples involving $2 times 2$ matrices showing that the last inequality is, in general, violated in case that neither (a) nor (b) is fulfilled.;


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