 Solution

Although $ab$ need not be self-adjoint, it is similar to a Positive element of a C-star algebra. Namely,By congruence preserves positivity in a C-star algebra, $d\ge 0$. In fact $b\ge 1$ givesso $d\ge a\ge 1$. Similar elements have the same spectrum of an element, because $s^{-1}(x-\lambda 1)s$ is invertible exactly when $x-\lambda 1$ is. Therefore the stronger conclusion isPositivity of the spectrum is being asserted, not positivity of the generally non-self-adjoint product itself.