Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces
Abstract
A maximal plurisubharmonic function on an infinite-dimensional Hilbert space can have an injective, noncompact Levi operator at every point. This raises the question of what replaces the finite-dimensional determinant criterion for maximality. We prove that a twice continuously differentiable plurisubharmonic function is maximal if its Levi operator is compact at every point, without any boundedness assumption on the function. Using the normalized determinants of Fujii and Seo, we also characterize local maximality by the condition that the infimum of the spectrum of the Levi operator is zero. The same condition characterizes maximality when the function is bounded on the closure of every bounded open set whose closure lies in the domain. The compactness result follows from a more general spectral criterion. Its proof combines a bounded transformation with the smooth variational principle of Borwein and Preiss to establish comparison on bounded open sets whose closures need not be compact.
Related
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BibTeX
@misc{ahag2026maximal,
title = {{Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces}},
author = {Per Åhag and Rafał Czyż and Antti Perälä and Jani Virtanen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.10742},
eprint = {2605.10742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.10742},
}
RIS
TY - UNPB
TI - Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces
AU - Per Åhag
AU - Rafał Czyż
AU - Antti Perälä
AU - Jani Virtanen
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2605.10742
UR - https://arxiv.org/abs/2605.10742
AB - A maximal plurisubharmonic function on an infinite-dimensional Hilbert space can have an injective, noncompact Levi operator at every point. This raises the question of what replaces the finite-dimensional determinant criterion for maximality. We prove that a twice continuously differentiable plurisubharmonic function is maximal if its Levi operator is compact at every point, without any boundedness assumption on the function. Using the normalized determinants of Fujii and Seo, we also characterize local maximality by the condition that the infimum of the spectrum of the Levi operator is zero. The same condition characterizes maximality when the function is bounded on the closure of every bounded open set whose closure lies in the domain. The compactness result follows from a more general spectral criterion. Its proof combines a bounded transformation with the smooth variational principle of Borwein and Preiss to establish comparison on bounded open sets whose closures need not be compact.
ER -