I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|_2 \leq 1.$

The argument I was using to show this is that for any two square matrices $A$ and $B,$ it is always the case that $\|AB\|_2 = \|BA\|_2.$ I thought I read that this equality does hold in a reputable source, but I don't have access to it right now and I was unsuccessful in reproducing a proof.

I know the eigenvalues of $AB$ and $BA$ are the same modulo possibly having different numbers of zeros, so I'm worried that I might have ``remembered'' something that isn't true!

Any references/counterexamples for either question?