On algebras of polynomially compact operators

In this paper, we find sufficient and necessary conditions for a triangularizable closed algebra of polynomially compact operators to be commutative modulo the radical. We also prove that an algebraic algebra of operators of a bounded degree on a Banach space is triangularizable under some mild additional conditions. As a special case we obtain a result stating that every algebraic algebra of operators of bounded degree is triangularizable whenever its commutators are nilpotent operators.     

On reducibility of semigroups of compact quasinilpotent operators

The following generalization of Lomonosov's invariant subspace theorem is proved. Let be a multiplicative semigroup of compact operators on a Banach space such that for every finite subset of , where denotes the Rota-Strang spectral radius. Then is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space , every semigroup of compact quasinilpotent operators on is reducible.

(B) For every complex Hilbert space , for every semigroup of compact quasinilpotent operators on , and for every finite subset of it holds that .

The question whether the assertion (A) is true was considered by Nordgren, Radjavi and Rosenthal in 1984, and it seems to be still open.