Direct Integrals of Strongly Irreducible Operators

Strongly irreducible operators can be considered as building blocks for bounded linear operators on complex separable Hilbert spaces. Many bounded linear operators can be written as direct sums of at most countably many strongly irreducible operators. In this paper, we show that a bounded linear operator A is similar to a direct integral of strongly irreducible operators if its commutant {A}' contains a bounded maximal abelian set of idempotents. We find that bounded linear operators which are similar to direct integrals of strongly irreducible operators form a dense subset of ℒ(ℋ) in the operator norm.