Jordan blocks and strong irreducibility

An operator is said to be strongly irreducible if its com- mutant has no nontrivial idempotent. This paper first shows that if an operator is not strongly irreducible then the set of idempotents in its commutant is either finite or uncountable. The second part of the paper focuses on the Jordan block which is a well-known class of irreducible operators, and determines when a Jordan block is strongly irreducible. This work is an interplay of operator theory and complex function theory.