Let $T$ be a bounded operator on a complex Hilbert space $H$. Let $\sigma_c(T)$ denote its continuous spectrum. It seems that $\sigma_c$ can in general be quite weird and not "continuous" in appearance at all. I would like to know if there is a way to systematically construct operators whose continuous spectrum is, say, a given finite set of isolated points.
More concretely:
Question: Is there an example of $T$ for which $\sigma_c(T)$ consists of two points?
Thoughts: When $T$ is normal, an isolated point in its spectrum must be an eigenvalue, so one must look at the situation when $T$ is not normal. The Volterra operator has $\sigma_c(T)=\{0\}$. I wonder if one can use this to construct an operator with $\sigma_c(T)$ being two points.
Consider the direct sum of two copies of $H$. Let your operator be the Volterra operator $T$ on the first copy and $I+T$ on the second. That is, if we write $H = H_1 \oplus H_2$ with $H_1$ and $H_2$ the two copies, take $(f,g) \mapsto (Tf, (I+T)g)$.