Simultaneous Orbit Vanishing and Scalar Hypercyclicity

Definitions and mathematical statement

Let $T$ be an operator on a Hilbert space $H$, using the usual bounded linear-operator interpretation. The scalar field, separability, and dimension assumptions are not supplied. With iterates indexed by ${\mathbb{N}}_0$, define the single-vector condition \[ \forall x\in H:\qquad\inf_{n\in{\mathbb{N}}_0}\|T^nx\|=0. \tag{1} \] This means that for every $x$ and every $\varepsilon>0$, some iterate makes $\|T^nx\|<\varepsilon$; the iterate may depend on $x$ and $\varepsilon$.

The source contains three questions:

1. Does (1) imply \[ \forall u,v\in H:\quad \inf_{n\in{\mathbb{N}}_0}\left(\|T^nu\|+\|T^nv\|\right)=0\ ? \] The same iterate must make both norms small; separate infima do not express the requested conclusion.
2. If both $T$ and $3T$ are hypercyclic, must $2T$ be hypercyclic? Hypercyclicity means existence of a vector with dense forward orbit in $H$. The vectors witnessing hypercyclicity of $T$ and $3T$ need not be the same. The scalar factors are exactly $1,3,2$ as supplied; no full interval of factors is substituted.
3. Does "the first condition" imply that $T$ is hypercyclic? The natural antecedent is (1), but the text does not explicitly disambiguate this phrase from the preceding simultaneous-smallness question. Under the literal reading (1), no extra dense-orbit, transitivity, nontriviality, or growth assumption is present.

The third clause does not exclude zero or contracting operators, which signals a possible omitted hypothesis if a nontrivial hypercyclicity problem was intended. This defect is recorded rather than repaired. The source also does not specify whether the scalar-multiple question is independent of (1) or inherits the opening setup beyond being an operator on a Hilbert space. No answers to the bundled questions or additional hypotheses are asserted in this statement draft.

Short English statement

The entry bundles simultaneous smallness of two orbits, hypercyclicity of the intermediate scalar multiple $2T$, and a claimed connection from orbit smallness to hypercyclicity. The first question requires one shared iterate; the last appears to lack hypotheses, and the scope of the bundled clauses needs checking.

Sources

• Local UnsolvedMath registry, permanent ID 30000575, code catalogue entry, "Simultaneous Orbit Vanishing and Scalar Hypercyclicity." Proposed year 2006; proposer unspecified.
• Mini-Workshop: Hypercyclicity and Linear Chaos (2007), Oberwolfach Reports, pp. 2227--2276, DOI 10.4171/owr/2006/37; https://doi.org/10.4171/owr/2006/37. The displayed report year 2007 differs from the DOI and proposed year 2006. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied verification says operator, infimum, norm, vector, and scalar-multiple notation was restored, retaining all three questions.
• Literature triage checked 2026-08-21: open. It generically calls the item an unresolved problem/conjecture/direction and reports no authoritative resolution in the supplied check. That label does not resolve the apparent missing hypotheses in the third clause. Created 2026-08-25; updated 2026-09-16. Source content and status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.