Intereting Posts

Existence of sequences converging to $\sup S$ and $\inf S$
$f(A \cap B)\subset f(A)\cap f(B)$, and otherwise?
twice differentiable functions
If $cos(A-B)+cos(B-C)+cos(C-A)=\frac {-3}{2}$, prove that $cosA+cosB+cosC=sinA+sinB+sinC=0$
Running average of a convex function is convex
Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$
Showing a Ring of endomorphisms is isomorphic to a Ring
Must every subset of $\mathbb R$ contain $2$ homeomorphic distinct open sets?
Locally or Globally Lipschitz-functions
Reference Books on Cryptography
Proving that every group of order $4$ is isomorphic to $\Bbb Z_4$ or $\Bbb Z^*_8$
A partition of vertices of a graph
Applications of model theory to analysis
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
if a complex function $f$ is real-differentiable, then $f$ or $\overline{f}$ are complex-differentiable

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual inner product. Let $S$ be the shift operator:

$$

(S a)_n = a_{n-1}.

$$

If there a linear operator $A:\ell^2\to\ell^2$ such that $S=e^A$? I really doubt there is, but I’m not sure.

Context: I’m thinking about the momentum operator as a generator of the translation operator in quantum mechanics. One could think of $\ell^2(\mathbb{Z})$ as a discrete version of a particle on the line.

- Norm for pointwise convergence
- Is it possible to write any bounded continuous function as a uniform limit of smooth functions
- If $(I-T)^{-1}$ exists, can it always be written in a series representation?
- A counter example of best approximation
- Orthogonal Projections in Hilbert space
- Distributions on manifolds

- A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete
- $L^1(μ)$ is finite dimensional if it is reflexive
- Proving that the Fourier Basis is complete for C(R/$2*\pi$ , C) with $L^2$ norm
- Sum of Closed Operators Closable?
- Fourier transform and non-standard calculus
- Space of bounded continuous functions is complete
- Weak convergence of a sequence of characteristic functions
- Do there exist bounded operators with unbounded inverses?
- Fourier transform in $L^p$
- True/False: Self-adjoint compact operator

$l^{2}(\mathbb{Z})$ 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}

$$

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 $e^A = S$.

- Beside transcendental or uncomputable numbers what other types of numbers are there?
- The Supremum and Bounded Functions
- Consecutive non square free numbers
- A ring with few invertible elements
- are two metrics with same compact sets topologically equivalent?
- How to integrate $\int\frac{\ln x\,dx}{x^2+2x+4}$
- how to prove the chain rule?
- Alternative Expected Value Proof
- How do I prove that $\sum_{k=1}^{b-1} = \frac{(a-1)(b-1)}{2}$?
- Lower semi-continuity of one dimensional Hausdorff measure under Hausdorff convergence
- In how many ways can we arrange 4 letters of the word “ENGINE”?
- uniform random point in triangle
- Is a “network topology'” a topological space?
- Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely
- Approximating continuous functions with polynomials