The Necessary and Sufficient Condition for Strong Irreducibility of Cowen-Douglas Operators

Abstract:

      In this note, we show that a Cowen-Douglas operator is strongly irreducible if and only if its commutant algebra mod its Jocobson radical is isomorphic to a closed subalgebra of $H^{\infty}(D)$, where $D$ is the open unit disk, and $H^{\infty}(D)$ denotes the collection of bounded holomorphic functions on $D$.