Spectrum of Rank 1 Operators

Question

Given $\psi$ and $\phi$ in a Hilbert space $H$, we let $T$ be the rank-1 operator such that $$T\varphi=<\psi,\varphi>\phi.$$ It is easy to find the eigenvalues of $T$, they are $0$ and $<\psi,\phi>$. It is a bit harder to find the spectrum of $T$, as well as the point spectrum, continuous spectrum, and remainder spectrum of $T$.
The point spectrum (i.e., the set of $x\in\mathbb{C}$ such that $T-x$ is not injective) is pretty straightforward, it is simply the eigenvalues by looking at the kernel of $T$.
The eigenvalues are also in the spectrum, but I'm having a hard time determining if there are any other points in it.
The continuous spectrum (i.e. the set of $x\in\mathbb{C}$ such that $T-x$ is injective and the image of $T-x$ is dense) definitely does not include the eigenvalues. I tried thinking about a sequence $f_n$ that converges to every $f$ in $H$, but I couldn't come up with anything. My gut feeling is that $T-x$ is dense for every $x$ except the eigenvalues, as the image of $T$ is the span of $\phi$ in $H$ and I think this is dense.
My argument above would make the remainder spectrum (i.e. the set of $x\in\mathbb{C}$ such that $T-x$ is injective and the image of $T-x$ is not dense) the empty set.
Any help would be appreciated.

Answer 1

Any finite rank operator is a compact operator, and it's a known result that the only points in the spectrum of a compact operator are the eigenvalues. See theorem 35.17 here for the general statement.