Using functional calculus of bounded functions is easy to see that the application of the function $t \to \frac{t-i}{t+i}$ to any self-adjoint operator $T$ gives a unitary operator $U$ whose spectrum does not contain the point 1.
Rudin states in chapter $13$ the converse: that any unitary operator whose spectrum does not contain $1$ is obtained in this way. However, I think this cannot be proved using bounded functions since the function that one would need to find such $T$ has to take the unitary circle minus $1$ to the real line and hence is not bounded.
Is Rudin wrong in this statement?