Scalar-Stable Hypercyclic Operators on Nuclear Fréchet Spaces

Definitions and mathematical statement

Let $E$ be the nuclear Fréchet space in the source, over its intended scalar field $\mathbb K$, usually ${\mathbb{R}}$ or ${\mathbb{C}}$. A Fréchet space is a complete metrizable locally convex Hausdorff vector space. One standard nuclearity formulation requires that for every continuous seminorm $p$ there exist a continuous seminorm $q\geq p$ for which the canonical map between the Banach completions $E_q\to E_p$ is nuclear; these are completions of $E/\ker q$ and $E/\ker p$, respectively. The source does not give a narrower class of nuclear spaces.

For a continuous linear map $S:E\to E$, a vector $x$ is hypercyclic when $\{S^nx:n\in{\mathbb{Z}}_{\geq0}\}$ is dense in $E$, and $S$ is hypercyclic if such a vector exists. The requested assertion, preserving the scalar quantifiers, is \[ \forall E\text{ in the intended class}\ \exists T\in\mathcal L(E) \ \forall\lambda\in\mathbb K\setminus\{0\}\ \exists x_\lambda\in E, \qquad \overline{\{\lambda^nT^nx_\lambda:n\geq0\}}=E. \] The same operator $T$ must work for every nonzero scalar. The hypercyclic vector may depend on $\lambda$; a common vector for all multiples is a stronger statement and is not requested. The case $\lambda=1$ includes hypercyclicity of $T$ itself. Scalars of arbitrary nonzero modulus are included, not merely scalars on the unit circle.

There is an essential space-scope omission: a dense orbit implies separability, and nonzero finite-dimensional real or complex spaces do not support hypercyclic linear operators in the usual topology, whereas finite-dimensional spaces are nuclear Fréchet spaces. The source says every nuclear Fréchet space without explicitly restricting to a separable infinite-dimensional setting. The literal broad quantifier therefore needs qualification; the draft records this issue rather than silently inserting the customary nontrivial hypotheses or treating the literal unrestricted version as a verified open conjecture. The treatment of the zero space and the scalar field are also unspecified.

The literature triage checked 2026-08-21 reports no authoritative resolution of the extracted claim located in its targeted search. The verification note says all quantifiers were preserved during notation cleanup; it does not supply the missing space restrictions. The open label remains dated source metadata, not an independently verified assertion about every space covered by the literal wording.

Short English statement

Seek one continuous linear operator whose every nonzero scalar multiple is hypercyclic; each multiple may have its own dense-orbit vector. The source omits separable infinite-dimensional restrictions, even though its literal nuclear Fréchet class includes finite-dimensional spaces. That scope issue is explicit rather than silently repaired.

Sources

• Local UnsolvedMath registry, permanent ID 30000574, source code catalogue entry, "Scalar-Stable Hypercyclic Operators on Nuclear Fréchet Spaces," including the original and cleaned all-scalar question and quantifier-verification note.
• 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. This is also the sole supplied literature reference. The citation uses 2007, while the DOI and proposed-year metadata use 2006.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.