Question about compact operator

Question

So here is my question,

Let $H$ be a Hilbert-space and $K:H\rightarrow H$ a compact operator. I know that if $K$ is self adjoint, then it has one eigenvalue $\mu$ such that $|\mu|=||K||$. Can some one give me a nice counter example of a not self adjoint compact operator such that such a eigenvalue does not exist?

Thanks in advance!

Answer 1

The standard example is the Volterra operator, e.g., on $L^2[0,1]$, given by $Vf(x)=\int_0^x f(t)\,dt$. It is compact, but not self-adjoint, and has no eigenvalues.