Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces
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| Format: | Preprint |
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2026
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| _version_ | 1866910209444675584 |
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| author | Åhag, Per Czyż, Rafał Perälä, Antti Virtanen, Jani |
| author_facet | Åhag, Per Czyż, Rafał Perälä, Antti Virtanen, Jani |
| contents | Let $H$ be a complex Hilbert space and let $Ω\subset H$ be a domain. In infinite dimensions, there is no canonical complex Monge--Ampère operator and no basis-free determinant of the Levi form. Hence, a determinant-type characterization of maximal plurisubharmonic functions is not immediate. We propose to use the normalized determinants of Fujii and Seo: for a bounded strictly positive operator $A$ and a unit vector $x\in H$, we set $Δ_x(A):=\exp\bigl(\langle (\log A)x,x\rangle\bigr)$, and we extend this naturally to non-invertible positive operators. We show that, for strictly positive operators, inequalities for $Δ_x$ precisely describe the chaotic order $\log A\ge \log B$, and we combine this observation with Kantorovich--Specht type bounds for positive operators.
For $u\in \mathcal{PSH}(Ω)\cap C^2(Ω)$ we define the \emph{Fujii--Seo determinant density} \[ \operatorname{FSD}(u)(a):=\inf_{\|x\|=1}Δ_x\!\bigl(D'D''u(a)\bigr),\qquad a\inΩ, \] and identify it with the lower spectral endpoint $\infσ(D'D''u(a))$. Thus, $\operatorname{FSD}(u)$ is precisely the infimum of the spectrum of the Levi form, and its vanishing gives a basis-independent criterion for pointwise degeneracy of the Levi form. We prove that maximality implies $\operatorname{FSD}(u)\equiv 0$, give sufficient global degeneracy criteria for maximality, and establish several comparison principles for $C^2$ plurisubharmonic functions, including results under uniform ellipticity bounds on the Levi form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10742 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces Åhag, Per Czyż, Rafał Perälä, Antti Virtanen, Jani Complex Variables Functional Analysis Primary 32U05, 46G05, Secondary 47A63, 47B65 Let $H$ be a complex Hilbert space and let $Ω\subset H$ be a domain. In infinite dimensions, there is no canonical complex Monge--Ampère operator and no basis-free determinant of the Levi form. Hence, a determinant-type characterization of maximal plurisubharmonic functions is not immediate. We propose to use the normalized determinants of Fujii and Seo: for a bounded strictly positive operator $A$ and a unit vector $x\in H$, we set $Δ_x(A):=\exp\bigl(\langle (\log A)x,x\rangle\bigr)$, and we extend this naturally to non-invertible positive operators. We show that, for strictly positive operators, inequalities for $Δ_x$ precisely describe the chaotic order $\log A\ge \log B$, and we combine this observation with Kantorovich--Specht type bounds for positive operators. For $u\in \mathcal{PSH}(Ω)\cap C^2(Ω)$ we define the \emph{Fujii--Seo determinant density} \[ \operatorname{FSD}(u)(a):=\inf_{\|x\|=1}Δ_x\!\bigl(D'D''u(a)\bigr),\qquad a\inΩ, \] and identify it with the lower spectral endpoint $\infσ(D'D''u(a))$. Thus, $\operatorname{FSD}(u)$ is precisely the infimum of the spectrum of the Levi form, and its vanishing gives a basis-independent criterion for pointwise degeneracy of the Levi form. We prove that maximality implies $\operatorname{FSD}(u)\equiv 0$, give sufficient global degeneracy criteria for maximality, and establish several comparison principles for $C^2$ plurisubharmonic functions, including results under uniform ellipticity bounds on the Levi form. |
| title | Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces |
| topic | Complex Variables Functional Analysis Primary 32U05, 46G05, Secondary 47A63, 47B65 |
| url | https://arxiv.org/abs/2605.10742 |