Spectrum of a unitary

Question

I have a unitary element $v$ in $C(S^{1}, \mathbb{C})$ with full spectrum (the whole circle). Is it possible to construct another unitary $u$ in $C(S^{1}, \mathbb{C})$ out of $v$ such that the spectrum of $u$ is not full (i.e. a unitary with smaller range)?

Answer 1

If I understand correctly, you have a surjective continuous map $v:S^1\to S^1$ and would like to construct a non-surjective continuous map $u:S^1\to S^1$ out of it. One way to do so is to let $u=f\circ v$ where $f:S^1\to S^1$ is continuous and not surjective. For example, $f(x+iy)=x+i|y|$ and $f(x+iy)=|x|+iy$ both do the job. So does a constant $f$, of course.

If you'd like to have some specific relation between $u$ and $v$, you should clarify the equation.