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Existence of hypercyclic algebras in Fréchet algebras
Abstract
We prove that every separable infinite-dimensional Fréchet algebra X admits a continuous linear operator T supporting a dense invariant hypercyclic algebra, giving an affirmative answer to a question of Bayart, Costa Jr. and Papathanasiou. In fact, every dense countable-dimensional subalgebra 𝒜 of X can be prescribed as an invariant hypercyclic algebra. When X admits a continuous norm, the operator can additionally be chosen in the form T=I+K, where K is nuclear, so that 𝒜 = span Orb(a,T) for any prescribed a∈𝒜backslash{0}.
Versions
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BibTeX
@misc{jr2026existence,
title = {{Existence of hypercyclic algebras in Fréchet algebras}},
author = {Fernando Costa Jr. and Álvaro Rocha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13983},
eprint = {2609.13983},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13983},
}
RIS
TY - UNPB
TI - Existence of hypercyclic algebras in Fréchet algebras
AU - Fernando Costa Jr.
AU - Álvaro Rocha
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.13983
UR - https://arxiv.org/abs/2609.13983
AB - We prove that every separable infinite-dimensional Fréchet algebra X admits a continuous linear operator T supporting a dense invariant hypercyclic algebra, giving an affirmative answer to a question of Bayart, Costa Jr. and Papathanasiou. In fact, every dense countable-dimensional subalgebra 𝒜 of X can be prescribed as an invariant hypercyclic algebra. When X admits a continuous norm, the operator can additionally be chosen in the form T=I+K, where K is nuclear, so that 𝒜 = span Orb(a,T) for any prescribed a∈𝒜backslash{0}.
ER -