Closed Common Hypercyclic Subspaces

Definitions and mathematical statement

Let $X$ be the topological vector space intended by the source, and let $(T_j)_{j\geq1}$ be a countable family of continuous linear operators $X\to X$. A vector $x$ is hypercyclic for $T_j$ if its forward orbit is dense: \[ \overline{\{T_j^n x:n\in{\mathbb{Z}}_{\geq0}\}}=X. \] Write $\operatorname{HC}(T_j)$ for this set and assume each $T_j$ is hypercyclic, meaning $\operatorname{HC}(T_j)\ne\varnothing$. A common hypercyclic vector belongs to $\bigcap_{j\geq1}\operatorname{HC}(T_j)$.

In the usual infinite-dimensional-subspace convention, the requested closed common hypercyclic subspace is a linear subspace $M\subseteq X$ such that \[ M\text{ is closed},\qquad\dim M=\infty,\qquad M\setminus\{0\}\subseteq\bigcap_{j\geq1}\operatorname{HC}(T_j). \] The zero vector is excluded from the hypercyclicity requirement. No invariance of $M$ under the operators is imposed.

The problem asks for conditions on the countable family and ambient space under which a common hypercyclic vector subspace can be chosen with these closedness and infinite-dimensionality properties. If an algebraic common hypercyclic subspace $N$ is already available, the target is existence of a suitable closed $M$; it does not assert that $\overline N$ automatically has the required property. The extract does not specify the ambient Banach or Fréchet class, scalar field, or any stronger density or invariance convention for the initial subspace.

Short English statement

Characterize when a countable family of hypercyclic operators has a closed infinite-dimensional linear subspace whose every nonzero vector is hypercyclic for every operator. Merely having common hypercyclic vectors or a nonclosed subspace does not state the requested closed-subspace property.

Sources

• Supplied report: Mini-Workshop: Hypercyclicity and Linear Chaos (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/37, pp. 2227--2276; proposed year 2006. https://doi.org/10.4171/owr/2006/37.
• Local UnsolvedMath registry, permanent ID 30000572, catalogue entry; created 25 August 2026, updated 16 September 2026. It labels the statement exact and retained byte-for-byte. The supplied literature triage of 21 August 2026 labels the extracted question open and reports no located authoritative resolution; it supplies no additional external reference. No independent literature assessment is asserted here.