Is the spectrum of a $T: L^2[0,a] \rightarrow L^2[0,a]$ real or complex?

Question

I was writing a proof for something else, but then this question came up. Does the spectrum of $T$ necessarily contain only real value? The reason I asked is because the $cl(\sigma) \cup cl(\rho) = \mathbb{C}$, so I think it is, but I am not sure. If it isn't, please give an example.

Thanks a lot.

Answer 1

The spectrum of $T$ could be any compact subset of $\mathbb C$. One way of verifying this is as follows: given $K\subset \mathbb C$, compact, fix an orthonormal basis $\{e_n\}$ of $L^2[0,a]$, and define $T$ by choosing a dense subset $\{q_k\}$ of $K$ and letting $$ Te_n=q_ne_n,\ \ \ \in n\in\mathbb N. $$ It is not hard to check that $\sigma(T)=K$.