Joint Spectral Radius, Operator Semigroups, and a Problem of W. Wojtyński

We investigate the connections between the invariant subspace problem for operator semigroups and the joint spectral radius. As a consequence, it is proved that any quasinilpotent Lie algebra of compact operators on a Banach space is triangularizable. We extend the Berger–Wang formula to precompact sets of essentially scalar operators and prove the continuity of the joint spectral radius on them.