Spectral and Finite-Dimensional Problems for Hypercyclic Tuples

Definitions and mathematical statement

Use the standard bounded commuting-tuple convention. For $T=(T_1,\ldots,T_n)$ on a topological vector space $X$, define \[ \operatorname{Orb}(T,x)=\{T_1^{k_1}\cdots T_n^{k_n}x:(k_1,\ldots,k_n)\in{\mathbb{Z}}_{\geq0}^n\}. \] The tuple is hypercyclic if some $x$ has $\overline{\operatorname{Orb}(T,x)}=X$. An operator $B$ is cyclic if some vector has dense linear span of $\{B^kx:k\geq0\}$; cyclicity is not the same as hypercyclicity. On a complex Banach space the point spectrum of its adjoint is $\sigma_p(B^*)=\{\lambda:\exists0\ne\ell\in X^*,\ B^*\ell=\lambda\ell\}$.

The source collects these separate questions. Given a hypercyclic tuple, does there exist a cyclic $B$ satisfying $BT_i=T_iB$ for every $i$? Can a commuting cyclic $B$ also be chosen with $\sigma_p(B^*)=\varnothing$? Whether this second clause is required for every input tuple or asks only for some such example is not explicit.

Does there exist a tuple and vector with a somewhere-dense but nondense orbit, meaning \[ \operatorname{int}_X\overline{\operatorname{Orb}(T,x)}\ne\varnothing, \qquad\overline{\operatorname{Orb}(T,x)}\ne X? \] Somewhere dense refers to the closure's interior, not the interior of the countable orbit itself.

Does there exist a hypercyclic $n$-tuple of matrices acting on ${\mathbb{C}}^n$, with the number of generators exactly the dimension? Separately, does there exist a hypercyclic $n$-tuple of hyponormal operators on an infinite-dimensional Hilbert space? Hyponormality means \[ T_i^*T_i-T_iT_i^*\geq0\qquad(1\leq i\leq n). \] The standard commuting assumption does not add double commutation with adjoints. These are existence questions for distinct classes of tuples, not a single tuple required to satisfy all properties simultaneously.

The source does not fix the general ambient Banach or Fréchet category, scalar field, tuple-length quantifier, or whether commutativity is part of its definition. The conventional orbit above must be matched to the report; a noncommuting word semigroup would be a different setting. In particular the adjoint-empty question needs an appropriate infinite-dimensional setting and cannot automatically be imposed in the finite-dimensional subquestion. The supplied literature triage checked 2026-08-21 reports no authoritative resolution of the extracted collection, but gives no separate current-status assessment for each clause.

Short English statement

The record asks about cyclic operators in tuple commutants, empty adjoint point spectrum, somewhere-dense nondense joint orbits, matrix tuples with length equal to dimension, and hyponormal hypercyclic tuples. Each clause has a distinct ambient setting and existence quantifier.

Sources

• Local UnsolvedMath registry, permanent ID 30000568, catalogue entry, Spectral and Finite-Dimensional Problems for Hypercyclic Tuples.
• Mini-Workshop: Hypercyclicity and Linear Chaos (2007), Oberwolfach Reports, pp. 2227--2276, DOI https://doi.org/10.4171/owr/2006/37.
• Supplied literature triage checked 2026-08-21 cites only that report; no clause-specific theorem, tuple-length range, or ambient-space convention is supplied.