{"paper":{"title":"Polynomial actions of unitary operators and idempotent ultrafilters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Mariusz Lema\\'nczyk, Stanis{\\l}aw Kasjan, Vitaly Bergelson","submitted_at":"2014-01-30T15:01:12Z","abstract_excerpt":"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\n  (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{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7869","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}