The similarity orbit of a normal operator

Abstract:

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