Vanishing Subsequences for Unbounded Operators

Definitions and mathematical statement

The exact extraction asks whether an unbounded operator $T$ can satisfy \[ \forall E_0\text{ infinite-dimensional closed}\ \forall(n_k)\ \exists x_0\in E_0:\quad T^{n_k}x_0\longrightarrow0. \] The literal linear-operator statement is vacuous: it does not require $x_0\ne0$, and $x_0=0$ belongs to every subspace and to every iterate domain of a linear operator, with $T^n0=0$. Thus this text cannot be promoted as a nontrivial open property without checking omitted hypotheses.

To expose a possible intended formulation, let $E$ be an infinite-dimensional Banach space and $T:D(T)\subset E\to E$ a linear operator, and put $D_\infty(T)=\bigcap_{m\geq0}D(T^m)$. A nontrivial candidate would ask for an operator satisfying \[ \forall E_0\subset E\text{ closed, infinite-dimensional}\ \forall(n_k)\text{ strictly increasing in }{\mathbb{Z}}_{\geq0}\ \exists x_0\in(E_0\cap D_\infty(T))\setminus\{0\}:\quad \|T^{n_k}x_0\|\to0. \] This candidate adds requirements not present in the source and is therefore not silently asserted as the recovered problem. The ambient space may instead be another topological-vector-space class; the source gives no domain, closedness, dense-definition or convergence topology. If arbitrary sequences include $n_k=0$ for all $k$, a nonzero-vector version is impossible, so the sequence convention is also material.

"Unbounded" must be defined: an unbounded densely defined operator, a discontinuous everywhere-defined linear map, and a bounded operator with unbounded powers are distinct notions. The excerpt selects none, and for an unbounded-domain interpretation every displayed iterate must be defined.

The supplied literature triage checked 21 August 2026 reports no authoritative resolution of the extracted claim, but does not resolve the zero-vector, sequence or domain gaps. Its open label is retained as metadata rather than a claim that the literal vacuous statement is an unresolved theorem.

Short English statement

As written, the property holds trivially by choosing the zero vector. A meaningful intended problem needs a nonzero vector, an unbounded-operator domain/topology and a sequence tending to infinity. Those requirements are not supplied, so the draft records the gap rather than silently repairing the conjecture.

Sources

• Local UnsolvedMath registry, permanent ID 30000569, exact source code catalogue entry; Vanishing Subsequences for Unbounded Operators. Proposed year 2006, status open. Corrected_verified editing only restores notation; it leaves $x_0\in E_0$ without a nonzero condition and "every sequence" without a growth condition.
• Supplied report citation: Mini-Workshop: Hypercyclicity and Linear Chaos (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/37, pp. 2227--2276, https://doi.org/10.4171/owr/2006/37. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2006/37, and reports no authoritative resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no precise corrected theorem is supplied.