SUMS OF STRONGLY IRREDUCIBLE OPERATORS

A bounded linear operator T on a complex Hilbert space H is irreducible if it has no reducing subspace other than the trivial ones (0) and H; it is strongly irre-ducible if every operator similar to T is irreducible. Equivalently, T is irreducible (resp. strongly irreducible) if the only projections (resp. idempotent operators) commuting with T are 0 and I. Since (strongly) irreducible operators can act only on a separable space, in the following we will restrict ourselves to operators on such spaces. Strongly irreducible operators were first considered by Gilfeather [16]. Jordan block