If $B^{1/2}A=AB^{1/2}$. It is true that $AB=BA$? Let $E$ be a complex Hilbert space. Let $A\in \mathcal{L}(\mathcal{H})$ and let $B\in \mathcal{L}(E)^+$.

Assume that $B^{1/2}A=AB^{1/2}$. It is true that $AB=BA$?

Thank you

$\def\h{^{1/2}}$
$$ AB=AB\h B\h = (B\h A) B\h= B\h (AB\h)=B\h B\h A=BA\,.$$
