I am looking for a proof of the following: \begin{equation*} |||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| \end{equation*} Where A, B are positive, hermitian matrices, and $|||⋅|||$ is a unitarily invariant norm.
Perhaps someone has a reference to a book, or paper that contains such a proof? I have search myself and cannot seem to find anything. Also tried writing matlab code to find counterexamples and prove the contrary but that does not return anything - which leads to me to believe a proof must exist. Thank you.
The inequality does not always hold. Put $X=A^{1/2}$ and $Y=B^{1/2}$. The inequality is equivalent to $|||X^2Y^2|||\le|||XY|||^2$ for every pair of positive definite matrices $X,Y$ and for every unitarily invariant matrix norm $|||\cdot|||$. Now, take $X=\pmatrix{2&0\\ 0&1},\ Y=\pmatrix{2&1\\ 1&1}$ and the operator norm (induced 2-norm), we have $$ |||X^2Y^2|||\approx23.6>21.8\approx|||XY|||^2. $$