The essential spectrum, norm, and spectral radius of abstract multiplication operators

Let $E$ be a complex Banach lattice and $T$ is an operator in the centrum $Z(E)=\{T: |T|\le \lambda I \mbox{ for some } \lambda\}$ of $E$. Then the essential norm $\|T\|_{e}$ of $T$ equals the essential spectral radius $r_{e}(T)$ of $T$. We also prove $r_{e}(T)=\max\{\|T_{A^{d}}\|, r_{e}(T_{A})\}$, where $T_{A}$ is the atomic part of $T$ and $T_{A^{d}}$ is the non-atomic part of $T$. Moreover $r_{e}(T_{A})=\limsup_{\mathcal F}\lambda_{a}$, where $\mathcal F$ is the Fr\'echet filter on the set $A$ of all positive atoms in $E$ of norm one and $\lambda_{a}$ is given by $T_{A}a=\lambda_{a}a$ for all $a\in A$.