Operators with 𝐶*-algebra generated by a unilateral shift

Abstract

If T T is an operator on a Hilbert space H \mathcal {H} , this paper gives necessary and sufficient conditions on T T such that C ∗ ( T ) {C^{\ast } }(T) , the C ∗ {C^{\ast } } -algebra generated by T T , is generated by a unilateral shift of some multiplicity. This result is then specialized to the cases in which T T is a hyponormal or subnormal operator. In particular, it is shown how to prove a recent conjecture of C. R. Putnam as a consequence of our result.