In a previous question, it is shown that the rightward shift $S e_i = e_{i + 1}$ is a normal operator (here $\{e_i\}$ are the standard basis for $\ell^2(\mathbb Z)$), and so falls under the purview of the continuous functional calculus of normal operators on a Hilbert space. In particular, a logarithm can be defined in this case.
Consider now the rightward shift $S$ defined on $\ell^2(\mathbb N)$. This operator is no longer normal, and so to define a logarithm one might think to use the analytic functional calculus on bounded linear operators:
$$\log S = \int_\gamma \frac{\log (\zeta)}{(\zeta - S)} d \zeta \, ,$$
where $\gamma$ is an appropriate contour. However, $0$ belongs to the spectrum of $S$, and so there is no branch cut from $0$ to $\infty$ disjoint from the spectrum.
Does this mean there is no hope? Is there a way to get around this problem, or is there a proof that $S$ admits no logarithm?