The essential spectral radius of dominated positive operators

Abstract:

Let $E$ be an ${\operatorname {AL} _p}$-space with $1 \leqslant p \leqslant \infty$. We prove that if a positive operator $S \in \mathcal {L}(E)$ satisfies the Doeblin conditions and $r(S) \leqslant 1$, then $S$ is quasi-compact, i.e., ${r_{\operatorname {ess} }}(S) < 1$. We then deduce the following result about the monotonicity of the essential spectral radius: Let $S,\;T \in \mathcal {L}(E)$ be such that $0 \leqslant S \leqslant T$. If $r(S) \leqslant 1$ and ${r_{\operatorname {ess} }}(T) < 1$, then ${r_{\operatorname {ess} }}(S) < 1$.