A note on majorization properties of the Lieb function

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Marek Niezgoda


In this note, the Lieb function $(A,B) \to \Phi (A,B) = \tr \exp ( A + \log B )$ for an Hermitian matrix $A$ and a positive definite matrix $B$ is studied. It is shown that $\Phi$ satisfies a majorization property of Sherman type induced by a doubly stochastic operator. The variant for commuting matrices is also considered. An interpretation is given for the case of the orthoprojection operator onto the space of block diagonal matrices.

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