On Banach lattices with Levi norms

Abstract:

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.