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Invariant subspaces for free linearizations of Lipschitz maps
Abstract
We consider the invariant subspace problem for operators induced by Lipschitz self-maps on Lipschitz-free spaces. Besides the usual free linearizations f̂ of basepoint-preserving Lipschitz self-maps, we consider a wider class of operators Tf,e which are naturally defined for arbitrary Lipschitz self-maps. We show, among other results, that every f̂ admits a non-trivial invariant subspace whenever the underlying metric space contains a compact ball, or has at least two connected components one of which has non-empty interior. We also obtain corresponding positive results for Tf,e under isolated-point, compact-ball and disconnectedness assumptions, and discuss some consequences for linear dynamics.
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BibTeX
@misc{coine2026invariant,
title = {{Invariant subspaces for free linearizations of Lipschitz maps}},
author = {Clément Coine and Pedro L. Kaufmann and Colin Petitjean and Sebastián Tapia-García},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.02213},
eprint = {2609.02213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02213},
}
RIS
TY - UNPB
TI - Invariant subspaces for free linearizations of Lipschitz maps
AU - Clément Coine
AU - Pedro L. Kaufmann
AU - Colin Petitjean
AU - Sebastián Tapia-García
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.02213
UR - https://arxiv.org/abs/2609.02213
AB - We consider the invariant subspace problem for operators induced by Lipschitz self-maps on Lipschitz-free spaces. Besides the usual free linearizations f̂ of basepoint-preserving Lipschitz self-maps, we consider a wider class of operators T_f,e which are naturally defined for arbitrary Lipschitz self-maps. We show, among other results, that every f̂ admits a non-trivial invariant subspace whenever the underlying metric space contains a compact ball, or has at least two connected components one of which has non-empty interior. We also obtain corresponding positive results for T_f,e under isolated-point, compact-ball and disconnectedness assumptions, and discuss some consequences for linear dynamics.
ER -