Jordan blocks and strong irreducibility (Q416761)

Let \(H\) be a complex, separable Hilbert space and let \(B(H)\) be the algebra of all bounded linear operators on \(H\). An operator \(T\in B(H)\) is called strongly irreducible if its commutant does not contain non-trivial idempotents. The paper under review contains two results related to this class of operators. First, it is shown that if \(T\) is not strongly irreducible, then the set of all idempotents in its commutant is either finite or uncountable. Next, it is shown that the Jordan block \(S(\theta)\) associated to an inner function is not strongly irreducible if and only if \(\theta\) admits a corona decomposition.