Pure Subnormal Operators Have Cyclic Adjoints

We shall prove that a pure subnormal operator has a cyclic adjoint. This answers a question raised by J. Deddens and W. Wogen in 1976. We first prove that a subnormal operatorSon a Hilbert space H has a cyclic adjoint if and only if there exists a compactly supported Borel measureμin the complex plane and a one-to-one linear mapA: H→L2(μ) such thatAS=NμAwhereNμ=MzonL2(μ) . Second, we show that for any pure subnormal operatorS, there exists a one-to-one map intertwiningSand a *-cyclic normal operator. This technique of intertwining is also used to give new proofs of some known results on the cyclicity of adjoints. An application of the main result shows that every pure subnormal operator has a matrix representation that is “almost” lower triangular.