  Using a factorization theorem of Douglas, we prove functional
characterizations of trace spaces $H^s(\partial \Omega)$ involving a family of
positive self-adjoint operators. Our method is based on the use of a suitable
operator by taking the trace on the boundary $\partial \Omega$ of a bounded
Lipschitz domain $\Omega \subset \mathbb R^d$ and applying Moore--Penrose
pseudo-inverse properties together with a special inner product on
$H^1(\Omega)$. Moreover, generalized results of the Moore--Penrose
pseudo-inverse are also established.
