Search

If N N is a bounded normal operator on a separable Hilbert space H \mathcal {H} , let S ( N ) \mathcal {S}(N) denote the similarity orbit of N N in L ( H ) L(\mathcal {H}) and let S k ( N ) {\mathcal {S}_k}(N) denote the set of all compact perturbations of elements of S ( N ) \mathcal {S}(N) . It is proved that S ( N ) ( S K ( N ) ) \mathcal {S}(N)({\mathcal {S}_K}(N)) is norm closed in L ( H ) L(\mathcal {H}) if and only if the spectrum (essential spectrum) of N N is finite. If the essential spectrum of N N is infinite and M M is a normal operator whose spectrum is connected and contains that of N N , then M M is in the closure of S ( N ) \mathcal {S}(N) . If the spectrum of N N is connected, this result characterizes the normal elements of the closure of S ( N ) \mathcal {S}(N) . A normal operator is similar to a nonquasidiagonal operator if and only if its essential spectrum contains more than two points.