Hypercyclic Subspaces on ω and a Partial Answer to a Question of Petersson

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.