Spectra of typical Hilbert space operators

Let B(H) B ( H ) denote the algebra of all bounded linear operators on a separable Hilbert space, equipped with the norm topology. A property is called typical if the set of operators fulfilling the property is co-meager. We show that having non-empty continuous spectrum is a typical property and that the set of operators with empty continuous spectrum is dense in B(H) B ( H ) . In addition, we show that the set of operators with empty point spectrum is nowhere dense. Moreover, we characterize the closure of the set of operators whose spectrum and point spectrum coincide.