Compactness and weak compactness of linearized Holomorphic-Lipschitz maps
Abstract
Any holomorphic Lipschitz map f, defined on the open unit ball BX of a complex Banach space X and taking values in a complex Banach space Y, can be uniquely and canonically associated with a bounded linear operator f̂ : 𝒢0(BX) ⟶ 𝒢0(Y) between the corresponding holomorphic Lipschitz-free spaces. The aim of this paper is to establish necessary and sufficient conditions for the compactness and weak compactness of such operators. In particular, we obtain a general characterisation of the compactness of f̂ in terms of metric properties of its symbol f. We then focus on the case where the target space is a subspace of ℓp, 1≤ p<∞. When the target space is ℓ1, we show that compactness and weak compactness of f̂ are both equivalent to f̂ not fixing a copy of ℓ1. Finally, when the target space is ℓp with 1<p<∞, we obtain a metric characterisation of the weak compactness of f̂. As a consequence, we show that compactness and weak compactness need not coincide, even when the target space is ℓp.
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BibTeX
@misc{bayart2026compactness,
title = {{Compactness and weak compactness of linearized Holomorphic-Lipschitz maps}},
author = {Frédéric Bayart and Mathis Lemay and Colin Petitjean},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26047},
eprint = {2610.26047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26047},
}
RIS
TY - UNPB
TI - Compactness and weak compactness of linearized Holomorphic-Lipschitz maps
AU - Frédéric Bayart
AU - Mathis Lemay
AU - Colin Petitjean
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26047
UR - https://arxiv.org/abs/2610.26047
AB - Any holomorphic Lipschitz map f, defined on the open unit ball B_X of a complex Banach space X and taking values in a complex Banach space Y, can be uniquely and canonically associated with a bounded linear operator f̂ : 𝒢_0(B_X) ⟶ 𝒢_0(Y) between the corresponding holomorphic Lipschitz-free spaces. The aim of this paper is to establish necessary and sufficient conditions for the compactness and weak compactness of such operators. In particular, we obtain a general characterisation of the compactness of f̂ in terms of metric properties of its symbol f. We then focus on the case where the target space is a subspace of ℓ_p, 1≤ p<∞. When the target space is ℓ_1, we show that compactness and weak compactness of f̂ are both equivalent to f̂ not fixing a copy of ℓ_1. Finally, when the target space is ℓ_p with 1<p<∞, we obtain a metric characterisation of the weak compactness of f̂. As a consequence, we show that compactness and weak compactness need not coincide, even when the target space is ℓ_p.
ER -