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 $a\in A$.
Categories: math.FA