# Do unitarily equivalent operators have the same spectrum?

First, some definitions:

We say that two bounded operators on Hilbert spaces $T:H_1\to H_1$ and $S:H_2\to H_2$ are unitarily equivalent if there is an unitary operator $U:H_1\to H_2$ such that $T=U^*SU$ (where the supscribed $*$ represent the Hilbert adjoint).

I'd already proved the following fact:

If $T$ and $S$ are unitarily equivalent self-adjoint operators, then they have the same spectrum.

Now... what if they are not self-adjoint? I was looking for some sufficient conditions (such as this) and, more important, counterexamples: two unitarily equivalent operators that don't share the same spectrum.

I tried to look for some simple examples (finite ranked and even compact infinite ranked), but had no success.

$$(U^*SU-\lambda I)= U^*(S-\lambda I)U$$ So $S-\lambda I$ is invertible iff $U^*SU-\lambda I$ is invertible, which leaves $\sigma(S)=\sigma(U^*SU)$.