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

Abstract

Let $E$ be a complex Banach lattice and $T$ is an operator in the centrum $Z(E)=\{T: |T|\le λI \mbox{ for some } λ\}$ of $E$. Then the essential norm $\Vert T{\Vert }_e$ of $T$ equals the essential spectral radius $r_e(T)$ of $T$. We also prove $r_e(T)=\max \{\Vert T_{A^d}\Vert ,r_e(T_A)\}$, where $T_A$ is the atomic part of $T$ and $T_{A^d}$ is the non-atomic part of $T$. Moreover $r_e(T_A)={\mathrm{limsup}}_{\mathcal{F}}{\lambda }_a$, where $\mathcal{F}$ is the Fréchet 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 $a\in A$.

arXiv commentLemma 1.9 and 1.10 of the 1980 paper [5] as stated are false. A slightly weaker version is true and the results of [5] and the current paper remain true using this weaker version. For details see the remarks after Theorem 2.1