Genericity of certain classes of unitary and self-adjoint operators

http://dx.doi.org/10.4153/CMB-1998-021-4
Canad. Math. Bull. 41(1998), 137-139

  Published:1998-06-01
 Printed: Jun 1998

In a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, form a dense $G_\delta$. A similar theorem for bounded self-adjoint operators with a given norm bound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon's result, with simplicity of the spectrum added.