{
  "schema_version": 1,
  "problem_number": "OWR-1323-008",
  "title": "Hypercyclic Subspaces on ω and a Partial Answer to a Question of Petersson",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In the report of the 2006 Oberwolfach mini-workshop on hypercyclicity, H. Petersson asked two questions about a (hereditarily) hypercyclic operator T on a separable Fréchet space X. First, does HC(T) ∪ {0} contain the range of an injective operator S ∈ L(X)? Second, for such an S, is every linearly independent n-tuple in Im S hypercyclic for T ⊕ ⋯ ⊕ T? We give a partial answer. Our main result concerns the space ω = K^ℕ. For every hypercyclic operator T on ω there is an injective S ∈ L(ω) such that (Sx_1, …, Sx_n) is hypercyclic for the n-fold direct sum of T whenever x_1, …, x_n are linearly independent. Its range is closed, so every hypercyclic operator on ω has a hypercyclic subspace. The proof uses a special case of a lemma of Shkarin (2011, Lemma 1.5). The hypercyclic-subspace consequence also follows quickly from that lemma combined with a criterion of Menet (2013, Theorem 4.8), and it answers, for an operator on ω and its iterates, a question raised by Menet. On separable Banach spaces, a folklore argument with the entire functional calculus answers the first question positively for every hypercyclic operator. For weakly mixing operators on spaces with a continuous norm, the source itself answers it. In its universal reading, for every such S, the second question has a negative answer: in all these settings there is an S as in the first question whose range contains a pair (x, Tx), and such a pair is never hypercyclic for T ⊕ T. In its existential reading, for some such S, it has a positive answer exactly for the weakly mixing operators, on ω and on spaces with a continuous norm. We leave the first question open for hypercyclic operators that are not weakly mixing on non-normable Fréchet spaces with a continuous norm, and for Fréchet spaces without a continuous norm other than ω. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.FA",
    "math.DS"
  ],
  "keywords": [
    "hypercyclic operator",
    "hypercyclic subspace",
    "space of all sequences",
    "operator range",
    "direct sums of operators",
    "strong operator topology",
    "linear dynamics",
    "Oberwolfach Reports",
    "OWR-1323-008",
    "math.FA",
    "math.DS",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1323-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-1323-008/paper.pdf?v=67463a00aa65",
  "doi": "10.5281/zenodo.23049792",
  "zenodo_record_url": "https://zenodo.org/records/23049792",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to Petersson's question (OWR 37/2006). The first question is answered yes on ω = K^ℕ for every hypercyclic operator (the hypercyclic-subspace consequence also follows from Shkarin 2011 with Menet 2013), on separable Banach spaces by a folklore argument, and in the source itself for weakly mixing operators on spaces with a continuous norm; the universal form of the second question is answered no. The first question stays open on Fréchet spaces without a continuous norm other than ω, and for non-weakly-mixing operators on non-normable spaces with a continuous norm. Unrefereed.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}