Weighted-Shift Supercyclicity Converse

Definitions and mathematical statement

For terminology, let $X$ be a topological vector space over $\mathbb K\in\{{\mathbb{R}},{\mathbb{C}}\}$, and let $T:X\to X$ be a continuous linear operator. For $A\subseteq\mathbb K$, call $T$ $A$-supercyclic if \[ \exists x\in X\setminus\{0\}\qquad \overline{\{aT^n x:a\in A,\ n\in{\mathbb{Z}}_{\geq 0}\}}=X. \] The closure is taken in the topology of $X$. Ordinary supercyclicity uses $A=\mathbb K$; ${\mathbb{R}}$-supercyclicity uses $A={\mathbb{R}}$, including when $X$ is complex. Hypercyclicity means that some $x\in X$ satisfies \[ \overline{\{T^n x:n\in{\mathbb{Z}}_{\geq 0}\}}=X. \] These definitions explain the terminology; they do not specify the missing source setting.

The supplied statement-verification field identifies the intended direction as passing from an associated weighted shift $S$ to a composition operator $C_f$. Here a composition operator has the form $C_f g=g\circ f$ on a function space preserved by composition with $f$. If $C_f:X\to X$ and its associated weighted shift $S:Y\to Y$ are defined on suitable topological vector spaces over the same scalar field, the reported questions have the schematic form \[ S\text{ is }\mathbb K\text{-supercyclic} \quad\Longrightarrow\quad C_f\text{ is }\mathbb K\text{-supercyclic}, \] and \[ S\text{ is }{\mathbb{R}}\text{-supercyclic} \quad\Longrightarrow\quad C_f\text{ is }{\mathbb{R}}\text{-supercyclic}. \] Neither implication is asserted here as a theorem or as a fully specified conjecture. The supplied record omits the spaces, their topologies, the transformation, the weights and indexing of the shift, the association between the operators, and the hypotheses under which the implications are intended. It therefore does not determine the domain of a universal quantifier over such pairs.

The surviving source fragment asks about "the other way around," supercyclicity, and ${\mathbb{R}}$-supercyclicity, but contains no complete mathematical assertion. The record labels the statement unrecoverable. Its supplied verification says that the official report asks whether supercyclicity of the associated weighted shift implies supercyclicity of the composition operator and immediately gives affirmative proved theorems. Separately, the supplied dated literature assessment reports converse comparisons for hypercyclicity and real and complex supercyclicity in a dissipative bounded-distortion setting, without reporting a general theorem for arbitrary factors. The definitions and hypotheses of that setting are absent. These are supplied source assessments, not independently verified mathematical conclusions; the exact intended theorem or remaining open question cannot be recovered from this record alone.

Short English statement

The record concerns whether supercyclicity, including supercyclicity using only real scalar multiples, passes from an associated weighted shift to a composition operator. The essential operator setting and hypotheses are missing. Supplied curation reports affirmative results under dissipative bounded-distortion hypotheses and labels the entry partially solved, while also marking its statement unrecoverable. A precise theorem or remaining open problem requires human review of the cited material.

Sources

• Local UnsolvedMath registry, permanent ID $30005898$, supplied complete source record titled Weighted-Shift Supercyclicity Converse; exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year $2024$, and supplies no individual proposer.
• Supplied source citation: Mini-Workshop: New Horizons in Linear Dynamics, Universality, and the Invariant Subspace Problem (2024), Oberwolfach Reports, pp. 1069--1096, DOI 10.4171/owr/2024/19. https://doi.org/10.4171/owr/2024/19. The surviving statement fragment also contains the page marker $1083$. The report is the supplied authoritative reference, but its relevant mathematical passage is not reproduced completely in the record and has not been independently consulted here.
• Supplied literature source: D'Aniello and Maiuriello, supercyclicity. https://arxiv.org/abs/2404.04028.
• Supplied literature source: D'Aniello, Darji and Maiuriello, shift-like operators. https://arxiv.org/abs/2107.12103.
• Supplied literature source: Oberwolfach Report 19/2024. https://doi.org/10.4171/owr/2024/19.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. It labels the status partially solved and reports converse factor comparisons for hypercyclicity and real and complex supercyclicity in the dissipative bounded-distortion setting, without a general theorem for arbitrary factors. The background explicitly calls this dated literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims.