Talk:Unilateral shift operator

Context needed

This article seems thin, misses any statement of connections to the theory of shift operators in general, and seems to get lost in the weeds from the word go. An overview should mention:

• Baker's map
• Frobenius-Perron operator
• Sofic shifts and shift space
• Composition operator
• Shift matrix

Relationships to C*-algebras and measure-preserving dynamical systems. Most importantly, an overview of "why are shift spaces interesting in the first place?" (answer: dynamical systems) linas (talk) 23:40, 3 April 2026 (UTC)

I put together an utterly slap-dash overview intro, trying to indicate that this topic is far far broader than what was currently written. For starters, the article could mention that the $\ell ^{2}$ space is the same as the Hilbert space of complex-valued functions on the unit interval viz $f(x)=\sum _{k}a_{k}\exp(i2\pi kx)$. This fourrier decomposition is deceiving, in that it obscures the interesting "period doubling" basis $\exp(i\pi 2^{n}(2l+1)x)$ commonly used to study the fractals that show up in dynamical systems. It should also explain why the "Forward shift" is called the Koopman operator, and why the backward shift is called the Frobenius-Perron operator. It should also mention that the eigenfunctions of the shift operators correspond to character (mathematics); this includes not only the Bernoulli polynomials, but the Dirichlet L-functions. See, for example, multiplication theorem, scroll down to the section called "Bernoulli map" for details.

Shift operators are a very, very rich topic, connected to many branches of mathematics. Roughly speaking, the unilateral shift operator, applied to measurable spaces, give you the measure-preserving dynamical systems. There must be tens of thousands papers on the topic. I wrote what I hoped would be the very simplest possible introduction to the topic (I suspect I failed), for a relatively simple special case, the beta shift. It is here: https://arxiv.org/abs/1812.10593 The first 20 pages cover shifts in this general setting. The bad news is that I vastly over-simplified; the beta shift is a 1D problem, after all. My apologies, but perhaps it gives a toe-hold for the proper full theory of shift operators, which can be found in various books from the golden age of chaos theory.

Here's another one, that's really cool: Clock and shift matrices. In QM, the shifts apparently carry the information, and the clocks, uhh... or something like that... well, I don't understand it. I don't see any articles on the infinite-dimensional aka shift operator/clock operator version; here, the shift operator is the quantum harmonic oscillator, and the clock part has something to do with the Bogoliubov transformation but I don't fully understand. Or something. This is the same Bogoliubov that gives you Hawking radiation, so getting the details worked out correctly would be ... interesting. Of course, not in this article, but generically, its an interesting, broad avenue. linas (talk) 02:55, 4 April 2026 (UTC)