Polynomial actions of unitary operators and idempotent ultrafilters

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 $\mathrm{d}\mathrm{e}\mathrm{g}P\le N$ 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 ${\cal H}=\bigoplus_{r\geq 1}{\cal H}^{(N)}r$intoclosed,$U$-invariant subspaces such that (a) for any polynomial $P\in {\mathcal{P}}_{\le N}$, we have $$p\text{-}\text{\hspace{0.17em}⁣}\underset{n\in \mathbb{N}}{\lim }{\left(U{\mathrm{\mid }}_{{\mathcal{H}}_r^{(N)}}\right)}^{P(n)}=0_{{\mathcal{H}}_r^{(N)}}orI{d}_{{\mathcal{H}}_r^{(N)}},foreachr\ge 1;$$ (b) for each $r\mathrm{\ne }s$ there exists $Q\in {\mathcal{P}}_{\le N}$ such that $$p\text{-}\text{\hspace{0.17em}⁣}\underset{n\in \mathbb{N}}{\lim }{\left(U{\mathrm{\mid }}_{{\mathcal{H}}_r^{(N)}}\right)}^{Q(n)}\mathrm{\ne }p\text{-}\text{\hspace{0.17em}⁣}\underset{n\in \mathbb{N}}{\lim }{\left(U{\mathrm{\mid }}_{{\mathcal{H}}_s^{(N)}}\right)}^{Q(n)}\mathrm{.}$$ 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 $$\text{label}ab1G=\{P\in {\mathcal{P}}_{\le N}:p\text{-}\text{\hspace{0.17em}⁣}\underset{n\in \mathbb{N}}{\lim }{U}^{P(n)}=Id\}$$ and $p,$\text{-}$!\lim{n\in$\mathbb{N}$} U ^{Q(n)}=0;;\mbox{for each};;Q\in{\cal P}{\leq N}\setminus G.$Themainresultofthepaperstatesthatasubgroup$G\subset {\cal P}{\leq N}$satisfying$\max{{\rm deg}, P::P\in G}=N$isan$N$-rigidity group if and only if $G$ has finite index in ${\mathcal{P}}_{\le N}$.