Is a unitary opertor with $\sigma(U)=\lbrace 1\rbrace$ necessarily be the identity?

Question

In a complex, infinite dimensional, separable Hilbert space, we have a unitary operator $U\in\mathcal{B}(H)$ with only one element in its spectrum: $\sigma(U)=\lbrace 1\rbrace$. Is it true, that $U=I$?

Thanks for your answers!

Answer 1

Since $U$ is a unitary operator, $U-I$ is normal, so using Gelfand's spectral radius formula: $$0\le\Vert U-I\Vert=r(U-I)=\max\lbrace\lambda~\vert~\lambda\in\sigma(U-I)\rbrace\le\max\lbrace\lambda~\vert~\lambda\in\sigma(U)-\sigma(I)\rbrace=0$$ This implies $U=I$.

(The $\sigma(U-I)\subset\sigma(U)-\sigma(I)$ relation holds because $U$ and $I$ are commuting elements of a C*-algebra.)