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

Abstract

Let EE be a complex Banach lattice and TT is an operator in the center Z(E)={T:∣T∣≤λIfor someλ}Z\left(E)=\left\{T:| T| \le \lambda I\hspace{0.33em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\lambda \right\} of EE. Then, the essential norm ‖T‖e\Vert T{\Vert }_{e} of TT equals the essential spectral radius re(T){r}_{e}\left(T) of TT. We also prove re(T)=max{‖TAd‖,re(TA)}{r}_{e}\left(T)=\max \left\{\Vert {T}_{}\hspace{-0.35em}{}_{{A}^{d}}\Vert ,{r}_{e}\left({T}_{A})\right\}, where TA{T}_{A} is the atomic part of TT and TAd{T}_{}\hspace{-0.35em}{}_{{A}^{d}} is the nonatomic part of TT. Moreover, re(TA)=limsupℱλa{r}_{e}\left({T}_{A})={\mathrm{limsup}}_{{\mathcal{ {\mathcal F} }}}{\lambda }_{a}, where ℱ{\mathcal{ {\mathcal F} }} is the Fréchet filter on the set AA of all positive atoms in EE of norm one and λa{\lambda }_{a} is given by TAa=λaa{T}_{A}a={\lambda }_{a}a for all a∈Aa\in A.

Keywords

• essential spectral radius
• essential norm
• multiplication operators
• primary: 47g10
• secondary: 47b65
• 47b34