Polynomial actions of unitary operators and idempotent ultrafilters

Abstract

Let $p$ be an idempotent ultrafilter over $\mathbb{N}$. For a positive integer $N$, let ${\mathcal{P}}_{\le N}$ denote the additive group of polynomials $P\in \mathbb{Z}[x]$ with $\mathrm{deg}P\le N$ and $P(0)=0$. Given a unitary operator $U$ on a Hilbert space $\mathcal{H}$, we prove, for each $N\ge 1$, the existence of a unique decomposition $\mathcal{H}={⨁}_{r\ge 1}{\mathcal{H}}_r^{(N)}$ into closed, $U$-invariant subspaces such that (a) for any polynomial $P\in {\mathcal{P}}_{\le N}$, we have $$ p\, \text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{P(n)}=0_{{\cal H}_r^{(N)}}\;\mbox{or}\; Id_{{\cal H}_r^{(N)}},\; \mbox{for each}\; r\geq1 ; $$ (b) for each $r\ne s$ there exists $Q\in {\mathcal{P}}_{\le N}$ such that $$p\text{-}\underset{n\in \mathbb{N}}{\lim }{\left(U{|}_{{\mathcal{H}}_r^{(N)}}\right)}^{Q(n)}\ne p\text{-}\underset{n\in \mathbb{N}}{\lim }{\left(U{|}_{{\mathcal{H}}_s^{(N)}}\right)}^{Q(n)}.$$ In connection with this result we introduce the notion of rigidity group. Namely, a subgroup $G\subset {\mathcal{P}}_{\le N}$ is called an $N$-rigidity group if there exist an idempotent ultrafilter $p$ over $\mathbb{N}$ and a unitary operator $U$ on a Hilbert space $\mathcal{H}$ such that $$\label{ab1} G=\{P\in{\cal P}_{\leq N}:\: p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{P(n)}=Id\}$$ and $p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{Q(n)}=0\;\;\mbox{for each}\;\;Q\in{\cal P}_{\leq N}\setminus G.$ The main result of the paper states that a subgroup $G\subset {\mathcal{P}}_{\le N}$ satisfying $\max \{\mathrm{deg}P:P\in G\}=N$ is an $N$-rigidity group if and only if $G$ has finite index in ${\mathcal{P}}_{\le N}$.