On Banach lattices with Levi norms

Schmidt proved that an operator $T$ from a Banach lattice $E$ into a Banach lattice $G$ with property $(P)$ is order bounded if and only if its adjoint is order bounded, and in this case $T$ satisfies $\left\Vert \left\vert T\right\vert \right\Vert =\left\Vert \left\vert T^{\prime }\right\vert \right\Vert$. In the present paper the result is generalized to Banach lattices with Levi-Fatou norm serving as range, and some characterizations of Banach lattices with a Levi norm are given. Moreover, some characterizations of Riesz spaces having property $(b)$ are also obtained.