Spectral conditions and reducibility of operator semigroups

Let sigma(T) and r(T) denote, respectively, the spectrum and spectral radius of the bounded operator T on a complex Hilbert space H. Let S be a multiplicative semigroup of operators on H. We say sigma (resp. r) is submultiplicative on S if sigma(ST) subset-or-equal-to sigma(S)sigma(T) = {st : s is-an-element-of sigma(S), t is-an-element-of sigma(T)} (resp. r(ST) less-than-or-equal-to r(s)r(T)) for all S, T is-an-element-of S.We say sigma (resp. r) is permutable on S if sigma(RST) = sigma(SRT) (resp. r(RST) = r(SRT)) for all R, S <T is-an-element-of S.This paper is concerned with the effect of these conditions on (simultaneous) reducibility and triangularization of semi-groups.