Let $A$ be a matrix in $SL_2(\mathbb R)$. Define the trace norm to be
$\|A\| = \sqrt{\mathrm{tr}(A^* A)}. $
Is it true that this norm satisfies some kind of multiplicative property; for example:
$\|AB\| \leq \|A\|\cdot\|B\|.$
Can someone give me a brief reference where basic properties of this norm are stated and proved?
Thanks.