On reducibility of semigroups of compact quasinilpotent operators

The following generalization of Lomonosov's invariant subspace theorem is proved. Let ${\mathcal S}$ be a multiplicative semigroup of compact operators on a Banach space such that $\hat {r} (S_1, \ldots , S_n) = 0$ for every finite subset $\{S_1, \ldots , S_n\}$ of ${\mathcal S}$, where $\hat {r}$ denotes the Rota-Strang spectral radius. Then ${\mathcal S}$ is reducible.

This result implies that the following assertions are equivalent:

(A) For each infinite-dimensional complex Hilbert space ${\mathcal H}$, every semigroup of compact quasinilpotent operators on ${\mathcal H}$ is reducible.

(B) For every complex Hilbert space ${\mathcal H}$, for every semigroup of compact quasinilpotent operators on ${\mathcal H}$, and for every finite subset $\{S_1, \ldots , S_n\}$ of ${\mathcal S}$ it holds that $\hat {r}(S_1, \ldots , S_n) = 0$.

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