On the submultiplicativity and subadditivity of the spectral and essential spectral radius (Q258961)

Consider two bounded linear operators \(A\) and \(B\) on a Banach space \(X\). If \(A\) and \(B\) commute, then it is well known that the spectral radius is submultiplicative and subadditive, i.e., we have \(r(AB) \leq r(A)r(B)\) and \(r(A+B) \leq r(A) + r(B)\). If, on the other hand, the Banach space \(X\) is ordered by a generating and normal cone \(X_+\) and if the operators \(A\) and \(B\) are both positive, then the above inequalities also hold in case that the commutator \(AB - BA\) is only negative instead of being \(0\) (see the corollary to Lemma~1 in [\textit{M.~Zima}, Czech. Math. J. 49, No.4, 835--841 (1999, Zbl 1008.47004)]). In the paper under review, the authors generalise this result in various respects. They replace the condition \(AB \leq BA\) with a more general condition involving a third operator \(C\) and they do not only consider operators on Banach spaces, but elements of semi-normed (ordered) algebras. The authors' general approach allows them to derive a variety of interesting consequences such as, for instance, estimates for the essential spectral radius of positive operators.