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Polynomial actions of unitary operators and idempotent ultrafilters

Let $p$ be an idempotent ultrafilter over $\mathbb{N}$. For a positive integer $N$, let ${\cal P}_{\leq N}$ denote the additive group of polynomials $P\in\mathbb{Z}[x]$ with ${\rm deg}\, P\leq N$ and $P(0)=0$. Given a unitary operator $U$ on a Hilbert space ${\cal H}$, we prove, for each $N\geq1$, the existence of a unique decomposition ${\cal H}=\bigoplus_{r\geq 1}{\cal H}^{(N)}_r$ into closed, $U$-invariant subspaces such that (a) for any polynomial $P\in{\cal P}_{\leq N}$, 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 $r\neq s$ there exists $Q\in{\cal P}_{\leq N}$ such that $$ p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{Q(n)}\neq p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_s^{(N)}}\right)^{Q(n)}. $$ In connection with this result we introduce the notion of rigidity group. Namely, a subgroup $G\subset {\cal P}_{\leq N}$ 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 $\cal H$ 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 $G\subset {\cal P}_{\leq N}$ satisfying $\max\{{\rm deg}\, P:\:P\in G\}=N$ is an $N$-rigidity group if and only if $G$ has finite index in ${\cal P}_{\leq N}$.