On the submultiplicativity and subadditivity of the spectral and essential spectral radius

Abstract

Suppose that $A$ and $B$ are positive operators on an ordered Banach space with a normal and generating cone satisfying $0\le AB\le BA$. It is known that, in this case, we have

$$r\left(AB\right)\le r\left(A\right)r\left(B\right)\mathrm{and}r(A+B)\le r\left(A\right)+r\left(B\right).$$ In this article we consider less restrictive assumptions on $A$ and $B$ which give us the same inequalities as above. Moreover, we also prove these inequalities in a more general setting of ordered algebras, and we consider related results in the Banach algebra setting. We apply our results to the essential spectral radius $r_{\mathrm{ess}}$ of AM-compact operators and prove the equality $r_{\mathrm{ess}}(AB+BA)=2r_{\mathrm{ess}}\left(AB\right)$ under reasonable assumptions.