Topological Properties of Paranormal Operators on Hilbert Space

Abstract

Let $B(H)$ be the set of all bounded endomorphisms (operators) on the complex Hilbert space $H.T \in B(H)$ is paranormal if $||{(T - zI)^{ - 1}}|| = 1/d(z,\sigma (T))$ for all $z \notin \sigma (T)$ where $d(z,\sigma (T))$ is the distance from $z$ to $\sigma (T)$, the spectrum of $T$. If $\mathcal {P}$ is the set of all paranormal operators on $H$, then $\mathcal {P}$ contains the normal operators, $\mathfrak {N}$, and the hyponormal operators; and $\mathcal {P}$ is contained in $\mathcal {L}$, the set of all $T \in B(H)$ such that the convex hull of $\sigma (T)$ equals the closure of the numerical range of $T$. Thus, $\mathfrak {N} \subseteq \mathcal {P} \subseteq \mathcal {L} \subseteq B(H)$. Give $B(H)$ the norm topology. The main results in this paper are (1) $\mathfrak {N},\mathcal {P}$, and $\mathcal {L}$ are nowhere dense subsets of $B(H)$ when $\dim H \geq 2$, (2) $\mathfrak {N},\mathcal {P}$, and $\mathcal {L}$ are arcwise connected and closed, and (3) $\mathfrak {N}$ is a nowhere dense subset of $\mathcal {P}$ when $\dim H = \infty$.