From Local to Global Triangularization

It is shown that a semigroup of Shattenp-class operators is simultaneously triangularizable if each pair of operators in the semigroup is triangularizable. Several sufficient conditions for triangularizability of semigroups are obtained as corollaries. A “block triangularization” theorem for algebras of compact operators is established, consequences of which include a number of necessary and sufficient conditions for triangularization of such algebras.