  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 ${\cal P}_{\leq N}$.
