Hello all,

My question regards the geometric mean (GM) of two positive matrices. The definition of the GM for two positive matrices $(A,B)$ is given by: $M_0(A,B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\frac{1}{2}}A^{\frac{1}{2}}$. Moreover, when $A$ and $B$ commute, this definition reduces to $M_0(A,B)=A^{\frac{1}{2}}B^{\frac{1}{2}}$. The GM is known to be jointly concave in the pair $(A,B)$.

My question regards the reduced structure, namely $M_0(A,B)=A^{\frac{1}{2}}B^{\frac{1}{2}}$, for the general case where $A$ and $B$ **do no necessarily commute**. Is $A^{\frac{1}{2}}B^{\frac{1}{2}}$ jointly concave in the pair $(A,B)$ for any two positive matrices?
If so, how can one proof this?
In not? Which additional conditions on $(A,B)$ one should assume (without assuming commutativity) in order for it to be jointly concave in the pair?

Thank you very much in advance!