Sur l'équation ∂̄ dans un espace de Hilbert correction et généralisation d'un résultat de P. Raboin
Abstract
In his seminal paper on the ∂̅-equation in infinite dimensions [Rab], P. Raboin establishes the existence of C∞ solutions on a subspace HT of a Hilbert space H, given a closed C∞ right-hand side F and a self-adjoint Hilbert-Schmidt correlation operator T. His proof relies on a lemma stated without proof; we demonstrate that this lemma is false in general within infinite-dimensional spaces by constructing a counterexample based on a theorem by Borwein and Fabian [BF]. We show that the conclusion of [Rab] remains unaffected, as it follows from an independent argument. We then present, with a complete proof, a generalization obtained in 1985: the C∞ regularity assumption on F can be relaxed to C1, and the assumption regarding the correlation operator can be extended to any injective nuclear operator.
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BibTeX
@misc{rapp2026léquation,
title = {{Sur l'équation ∂̄ dans un espace de Hilbert correction et généralisation d'un résultat de P. Raboin}},
author = {Alain Michel Didier Rapp},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05799614v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sur l'équation ∂̄ dans un espace de Hilbert correction et généralisation d'un résultat de P. Raboin
AU - Alain Michel Didier Rapp
PY - 2026
DA - 2026/10/14
PB - HAL
UR - https://hal.science/hal-05799614v1
AB - In his seminal paper on the ∂̅-equation in infinite dimensions [Rab], P. Raboin establishes the existence of C^∞ solutions on a subspace H_T of a Hilbert space H, given a closed C^∞ right-hand side F and a self-adjoint Hilbert-Schmidt correlation operator T. His proof relies on a lemma stated without proof; we demonstrate that this lemma is false in general within infinite-dimensional spaces by constructing a counterexample based on a theorem by Borwein and Fabian [BF]. We show that the conclusion of [Rab] remains unaffected, as it follows from an independent argument. We then present, with a complete proof, a generalization obtained in 1985: the C^∞ regularity assumption on F can be relaxed to C^1, and the assumption regarding the correlation operator can be extended to any injective nuclear operator.
ER -