Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert Space, and assume that $T: H \to H$ is a possibly unbounded linear operator whose domain $D(T)$ is a dense subspace of $H$.
As usual, we define the resolvent set $\rho(T) \subseteq \mathbb{C}$ to be those $z \in \mathbb{C}$ such that $(T -z )^{-1} : H \to D(T)$ exists and is bounded with respect to the norm induced by the inner product $\langle \cdot, \cdot \rangle$. It is true that $\rho(T)$ is an open subset of $\mathbb{C}$.
Under what conditions on the operator $T$ is $\rho(T)$ known to be a connected set? Is this a fairly well-understood question, or are there difficulties involved?
I haven't thought about this question before, and don't know exactly how to approach it. I am familiar with some tools from functional analysis, such as the version of the spectral theorem developed via resolutions of the identity (e.g., from Rudin's book).
The cleanest example I have in my mind is $- \Delta : H^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, whose spectrum is $[0, \infty)$, and hence has connected resolvent set.
Any insight or reference suggestions are greatly appreciated!