$σ(f(T))$ not equal to $f(σ(T))?$

Question

I have no idea how to work it out and couldn't find any clue. Sorry if it's too trivial.

Let T be a bounded and non-normal operator, f(z)=zz' defined complex space (z' is a complex conjugate here).

What can be an example of such a T, that spectrum σ(f(T)) is not equal to f(σ(T))? As far as I checked, any simple finite matrix operator doesn't work here. Maybe some exponential?

Edit: corrected descriptions.

Answer 1

Let $S$ be the shift operator acting on $\ell^2\mathbb{N})$ by $Se_{n+1}=e_n$ and $Se_1=0.$ Then $SS^*= I$ and $\sigma(S)=\{|z|\le 1\}.$ Thus $\sigma(f(S))=\{1\}$ and $f(\sigma(S))=[0,1].$