Pure Subnormal Operators have Cyclic Adjoints

Abstract. 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 flrst prove that a subnormal operator S on a Hilbert space H has a cyclic adjoint if and only if there exists a compactly supported Borel measure „ in the complex plane and a one-to-one linear map A: H! L2(„) such that AS = N„A where N „ = Mz on L2(„). Second, we show that for any pure subnormal operator S, there exists a one-to-one map intertwining S and 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. J. Functional Analysis 162 (1999), no. 2, 379- 399. 1