Standard triangularization of semigroups of non-negative operators

We show that, under mild conditions, a semigroup of non-negative operators on L^p(X,μ) (for 1?p<∞) of the form scalar plus compact is triangularizable via standard subspaces if and only if each operator in the semigroup is individually triangularizable via standard subspaces. Also, in the case of operators of the form identity plus trace class we show that triangularizability via standard subspaces is equivalent to the submultiplicativity of a certain function on the semigroup.