Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Solution 1:

$l^{2}(\mathbb{Z})$ becomes $L^{2}[-\pi,\pi]$ under the unitary Fourier map $U : l^{2}(\mathbb{Z})\rightarrow L^{2}[-\pi,\pi]$ given by $$ \begin{align} U\{ a_{n}\}_{n=-\infty}^{\infty} & = \sum_{n=-\infty}^{\infty} a_{n}e^{in\theta},\\ U^{-1}f & = \{ (f,e^{in\theta})_{L^{2}[-\pi,\pi]}\}_{n=-\infty}^{\infty}. \end{align} $$ (The the inner-product $(\cdot,\cdot)_{L^{2}[-\pi,\pi]}$ on $L^{2}[0,2\pi]$ is normalized so that $\|1\|_{L^{2}[-\pi,\pi]}=1$.) The shift $S$ on $l^{2}(\mathbb{Z})$ becomes multiplication by $e^{i\theta}$ on $L^{2}[-\pi,\pi]$. That is, $S=U^{-1}EU$, where $$ (Ef)(\theta)=e^{i\theta}f(\theta). $$

The 'log' operator $L : L^{2}[-\pi,\pi]\rightarrow L^{2}[-\pi,\pi]$ defined by $(Lf)(\theta)=i\theta f(\theta)$ is a bounded normal linear operator such that $e^{L}=E=USU^{-1}$. So $$ S=U^{-1}e^{L}U=e^{U^{-1}LU}=e^{A},\;\;\; A = U^{-1}LU. $$ Sanity check: The spectrum of $L$ is $\{ i\theta : -\pi \le \theta \le \pi \}$ so that the spectrum of $e^{L}$ is the entire unit circle, as expected. It is possible to determine the explicit form for $A$ on $L^{2}(\mathbb{Z})$ from $$ \{ a_{n}\}_{n=-\infty}^{\infty} \mapsto \sum_{n}a_{n}e^{in\theta} \mapsto \left\{ \frac{1}{2\pi}\int_{-\pi}^{\pi} i\theta\sum_{n}a_{n}e^{in\theta}e^{-im\theta}\,d\theta\right\}_{m=-\infty}^{\infty} $$

Solution 2:

I think the answer is yes, by the spectral theorem, but $A$ does not admit an easy description. Since $S$ is a normal operator it admits a representation as $S = \int_{S^1} \lambda\,d\pi(\lambda)$ for some projection-valued measure $\pi$. One can now just take $L:S^1\to i\mathbf{R}$ to be any bounded Borel branch of the logarithm and put $A = \int_{S^1} L(\lambda)\,d\pi(\lambda).$

The above is a bit abstract and technical, and it's helpful at least for me to think about a finite-dimensional analogue. Let $S:\ell^2(\mathbf{Z}/N\mathbf{Z})\to\ell^2(\mathbf{Z}/N\mathbf{Z})$ be the operator defined by $(Sa)_n = a_{n+1\bmod{N}}$. Then $S$ has eigenvector $v_\zeta = (1,\zeta,\dots,\zeta^{N-1})$ with eigenvalue $\zeta$ for each $N$th root of unity $\zeta$. Since the vectors $v_\zeta$ are a basis for $\ell^2(\mathbf{Z}/N\mathbf{Z})$ there is an operator $A$ mapping each $v_\zeta$ to $L(\zeta)v_\zeta$, and this $A$ satisfies $e^A = S$.