Consider $\ell^2(\mathbb{Z})$ with standard basis $\{e_n\}_{n\in\mathbb{Z}}$ and the bilateral (right)-shift: $U(e_n)=e_{n+1}$. The spectrum of $U$ is the unit circle in the complex plane $\mathbb{C}$. There exists a unique self-adjoint bounded linear operator $T:\ell^2(\mathbb{Z})\to\ell^2(\mathbb{Z})$ such that $U = \exp(i T)$. Evidently, $T=-i\log(U)$ and the branch cut at $-1$ can be understood in Borel functional calculus (using monotone pointwise limits of continuous functions). What is $Te_0$? More generally, what is $T e_n$ for any $n\in\mathbb{Z}$?
Ref: Gert Pedersen, Analysis Now, GTM 118, Springer. E 4.5.1, E 4.5.5,