arxiv: 2605.10742 · v1 · submitted 2026-05-11 · 🧮 math.CV · math.FA

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Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces

Per {\AA}hag , Rafa{\l} Czy\.z , Antti Per\"al\"a , Jani Virtanen

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Pith reviewed 2026-05-12 04:06 UTC · model grok-4.3

classification 🧮 math.CV math.FA

keywords plurisubharmonic functionsmaximal plurisubharmonic functionsFujii-Seo determinantsLevi formHilbert spacesinfinite-dimensional complex analysiscomparison principles

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The pith

The Fujii-Seo determinant density of a C² plurisubharmonic function equals the lowest spectral value of its Levi form and vanishes when the function is maximal.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Domains in complex Hilbert spaces lack a canonical determinant for the Levi form of a plurisubharmonic function, so maximality has no obvious determinant-type test. The paper introduces the Fujii-Seo determinant density as the infimum, over all unit vectors, of the normalized exponential of the log of the Levi form operator at each point. This quantity is shown to coincide with the infimum of the spectrum of the Levi form. Maximality of the function forces the density to be zero everywhere, and the authors supply sufficient conditions under which global vanishing of the density implies maximality along with comparison principles that hold when the Levi forms satisfy uniform ellipticity bounds.

Core claim

For u in PSH(Ω) ∩ C²(Ω) the Fujii-Seo determinant density FSD(u)(a) is defined as the infimum over unit vectors of Δ_x applied to the Levi form D'D''u(a) and equals the lower endpoint of the spectrum of that operator; maximality of u implies FSD(u) ≡ 0, while global degeneracy of the Levi form supplies a sufficient criterion for maximality.

What carries the argument

The Fujii-Seo determinant density FSD(u), obtained by taking the infimum over unit directions of the normalized determinant Δ_x of the Levi form operator and identified with the infimum of its spectrum, which supplies a basis-independent test for pointwise degeneracy.

If this is right

Maximal C² plurisubharmonic functions have Levi forms whose spectrum reaches zero at every point.
Global vanishing of the lower spectral endpoint of the Levi form is sufficient for the function to be maximal.
Comparison principles between C² plurisubharmonic functions hold under uniform ellipticity bounds on their Levi forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The density offers a coordinate-free way to monitor degeneracy that may be useful for numerical schemes in infinite-dimensional pluripotential theory.
The same spectral-infimum construction could be applied to other operator-valued forms arising in infinite-dimensional complex geometry.
It suggests that maximality criteria in Banach spaces might be formulated by replacing determinants with spectral endpoints.

Load-bearing premise

The natural extension of the Fujii-Seo determinant from strictly positive operators to non-invertible positive operators preserves the inequalities that characterize the chaotic order.

What would settle it

A C² plurisubharmonic function that is maximal but whose Levi form has strictly positive lower spectral bound at some interior point would show that maximality does not force the Fujii-Seo density to vanish.

read the original abstract

Let $H$ be a complex Hilbert space and let $\Omega\subset H$ be a domain. In infinite dimensions, there is no canonical complex Monge--Amp\`ere 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 $\Delta_x(A):=\exp\bigl(\langle (\log A)x,x\rangle\bigr)$, and we extend this naturally to non-invertible positive operators. We show that, for strictly positive operators, inequalities for $\Delta_x$ precisely describe the chaotic order $\log A\ge \log B$, and we combine this observation with Kantorovich--Specht type bounds for positive operators. For $u\in \mathcal{PSH}(\Omega)\cap C^2(\Omega)$ we define the \emph{Fujii--Seo determinant density} \[ \operatorname{FSD}(u)(a):=\inf_{\|x\|=1}\Delta_x\!\bigl(D'D''u(a)\bigr),\qquad a\in\Omega, \] and identify it with the lower spectral endpoint $\inf\sigma(D'D''u(a))$. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a Fujii-Seo determinant density for the Levi form that gives a basis-free way to characterize maximality of plurisubharmonic functions in Hilbert spaces.

read the letter

The one thing to know is that the authors define a Fujii-Seo determinant density on the Levi form of C^2 plurisubharmonic functions and use its vanishing to characterize maximality in Hilbert space domains, along with some comparison principles. They take the normalized determinant Delta_x from Fujii and Seo, extend it to semidefinite operators, and set FSD(u)(a) to the inf over unit x of Delta_x on the Levi form at a. This turns out to be the infimum of the spectrum, giving a clean, basis-free way to measure degeneracy. From there they prove that maximal functions have FSD identically zero and obtain comparison results when the Levi form satisfies uniform ellipticity bounds. They also give sufficient conditions for maximality based on global degeneracy. This is new in the sense that it supplies the missing determinant-type tool for infinite dimensions. The paper does well by staying close to existing operator inequalities without inventing new machinery. The spectral identification follows from standard variational arguments once the extension is in place. Soft spots are small. The natural extension to non-invertible operators needs to be verified to preserve the chaotic order inequalities fully, though the stress test indicates it does. The comparison principles under ellipticity are stated but their sharpness or range of applicability would benefit from examples in the paper. No circularity or fitting issues show up. This is aimed at complex analysts and potential theorists working in infinite dimensions. Anyone needing a concrete handle on maximal functions there would get something out of it. It is worth sending to a serious referee, as the program is coherent and the results are checkable. I recommend proceeding with peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a theory of maximal plurisubharmonic functions in domains of complex Hilbert spaces using Fujii-Seo determinants. For u in PSH(Omega) cap C^2(Omega), it defines FSD(u)(a) as inf over ||x||=1 of Delta_x (D'D''u(a)), identifies this with inf sigma of the Levi form, proves maximality implies FSD(u) equiv 0, provides global degeneracy criteria, and establishes comparison principles including under uniform ellipticity.

Significance. This work addresses the lack of a canonical Monge-Ampere operator in infinite dimensions by providing a determinant-type characterization of maximality via the vanishing of the Fujii-Seo density, which coincides with the bottom of the spectrum of the Levi form. The approach leverages operator inequalities and chaotic order, offering potential applications in infinite-dimensional complex geometry. Strengths include the parameter-free nature of the spectral identification and the basis-independent criterion.

major comments (2)

[§2 (Fujii-Seo determinants for positive operators)] §2 (Fujii-Seo determinants for positive operators): The extension of the Fujii-Seo determinant Δ_x to non-invertible positive operators is described as 'natural' but requires a precise definition (e.g., how <(log A)x, x> is defined when 0 is in the spectrum). This is load-bearing for the claim that inequalities for Δ_x describe chaotic order log A ≥ log B, and thus for identifying FSD with inf σ.
[§4 (Maximality and FSD)] §4 (Maximality and FSD): The proof that maximality implies FSD(u) ≡ 0 should explicitly reference which comparison principle or degeneracy criterion is used, as the abstract combines this with Kantorovich-Specht bounds; clarify if the implication is direct from the spectral characterization or requires additional steps.

minor comments (2)

[Introduction] The notation for the Levi form D'D''u could be introduced with a brief reminder of its definition in the Hilbert space setting for accessibility.
[Throughout] Ensure all references to external results (e.g., Kantorovich-Specht type bounds) include precise citations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which will help improve the clarity of the presentation. We address each major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

Referee: [§2 (Fujii-Seo determinants for positive operators)] §2 (Fujii-Seo determinants for positive operators): The extension of the Fujii-Seo determinant Δ_x to non-invertible positive operators is described as 'natural' but requires a precise definition (e.g., how <(log A)x, x> is defined when 0 is in the spectrum). This is load-bearing for the claim that inequalities for Δ_x describe chaotic order log A ≥ log B, and thus for identifying FSD with inf σ.

Authors: We agree that an explicit definition strengthens the rigor of the extension. In the revised manuscript, we will define the extension of Δ_x to (possibly non-invertible) positive operators A via the spectral theorem: ⟨(log A)x, x⟩ := ∫_{[0,∞)} log(λ) d⟨E_A(λ)x, x⟩, where log(0) := -∞ by convention. This yields Δ_x(A) = 0 whenever the spectral measure charges 0. With this definition, the inequalities Δ_x(A) ≥ Δ_x(B) continue to characterize the chaotic order log A ≥ log B for positive operators, and the identification FSD(u) = inf σ(D'D''u) remains valid. We will insert this precise statement at the beginning of §2. revision: yes
Referee: [§4 (Maximality and FSD)] §4 (Maximality and FSD): The proof that maximality implies FSD(u) ≡ 0 should explicitly reference which comparison principle or degeneracy criterion is used, as the abstract combines this with Kantorovich-Specht bounds; clarify if the implication is direct from the spectral characterization or requires additional steps.

Authors: We appreciate the request for greater explicitness. The argument in §4 proceeds directly from the spectral identification FSD(u)(a) = inf σ(D'D''u(a)) together with the global degeneracy criterion (Theorem 3.4) that any maximal plurisubharmonic function must have inf σ(D'D''u) = 0 at every point; the latter is established via the comparison principles of §3 (which do employ Kantorovich–Specht bounds). No additional steps beyond this identification and the cited degeneracy criterion are required. In the revision we will add an explicit cross-reference to Theorem 3.4 and a short sentence clarifying the logical dependence at the beginning of the proof in §4. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on independent definitions and external bounds

full rationale

The paper defines the Fujii-Seo determinant density FSD(u) directly from the existing Fujii-Seo quantities Delta_x applied to the Levi form D'D''u of C^2 plurisubharmonic functions, then proves (rather than assumes) that this equals the spectral infimum inf sigma(D'D''u) and that maximality forces FSD ≡ 0. These proofs draw on the independently established notion of maximality for PSH functions, the chaotic order characterization for positive operators (shown via inequalities for Delta_x), and external Kantorovich-Specht bounds; no claimed result reduces by construction to a fitted input, self-referential definition, or load-bearing self-citation chain within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The central claim rests on standard properties of complex Hilbert spaces and positive operators together with one ad-hoc extension of the Fujii-Seo determinant; no numerical free parameters are introduced.

axioms (3)

standard math H is a complex Hilbert space and Omega subset H is a domain
Invoked in the opening sentence to set the ambient space for all definitions and results.
domain assumption For C^2 plurisubharmonic u the Levi form D'D''u(a) is a bounded positive operator on H
Required for the definition of FSD(u)(a) and for the application of Delta_x.
ad hoc to paper The Fujii-Seo determinant extends naturally to non-invertible positive operators while preserving the inequalities that characterize chaotic order
Stated explicitly in the abstract as the basis for the subsequent inequalities and identification.

invented entities (1)

Fujii-Seo determinant density FSD(u) no independent evidence
purpose: Basis-independent measure of pointwise degeneracy of the Levi form that vanishes for maximal functions
Newly defined as inf over unit vectors of Delta_x applied to the Levi form; no independent falsifiable handle outside the paper is supplied.

pith-pipeline@v0.9.0 · 5650 in / 1823 out tokens · 32984 ms · 2026-05-12T04:06:57.413359+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

P. M. Anselone and T. W. Palmer,Collectively compact sets of linear operators, Pacific J. Math.25(1968), 417–422

work page 1968
[2]

W. B. Arveson,Analyticity in operator algebras, Amer. J. Math.89(1967), 578–642

work page 1967
[3]

J. B. Conway and B. B. Morrel,Roots and logarithms of bounded operators on Hilbert space, J. Funct. Anal.70(1987), no. 1, 171–193

work page 1987
[4]

Cegrell,The general definition of the complex Monge–Ampère operator, Ann

U. Cegrell,The general definition of the complex Monge–Ampère operator, Ann. Inst. Fourier (Grenoble)54(2004), no. 1, 159–179

work page 2004
[5]

Cegrell,Maximal plurisubharmonic functions, Uzbek Math

U. Cegrell,Maximal plurisubharmonic functions, Uzbek Math. J. (2009), no. 1, 10–16

work page 2009
[6]

Dineen and F

S. Dineen and F. Gaughran,Maximal plurisubharmonic functions on domains in Banach spaces, Extracta Math.8(1993), no. 1, 54–60

work page 1993
[7]

S. S. Dragomir,Inequalities for the normalized determinant of positive operators in Hilbert spaces via some inequalities in terms of Kantorovich ratio, Math. Morav.28(2024), no. 2, 129–140

work page 2024
[8]

J. I. Fujii, S. Izumino and Y. Seo,Determinant for positive operators and Specht’s theorem, Sci. Math.1(1998), no. 3, 307–310

work page 1998
[9]

Fuglede and R

B. Fuglede and R. V. Kadison,On determinants and a property of the trace in finite factors, Proc. Natl. Acad. Sci. USA37(1951), 425–431

work page 1951
[10]

Fuglede and R

B. Fuglede and R. V. Kadison,Determinant theory in finite factors, Ann. of Math. (2)55 (1952), 520–530

work page 1952
[11]

J. I. Fujii and Y. Seo,Determinant for positive operators, Sci. Math.1(1998), no. 2, 153–156

work page 1998
[12]

J. I. Fujii and Y. Seo,Characterizations of chaotic order associated with the Mond–Shisha difference, Math. Inequal. Appl.5(2002), 725–734

work page 2002
[13]

Fredholm,Sur une classe d’équations fonctionnelles, Acta Math.27(1903), 365–390

I. Fredholm,Sur une classe d’équations fonctionnelles, Acta Math.27(1903), 365–390

work page 1903
[14]

Furuta,Results underlogA≥logBcan be derived from ones underA≥B≥0by Uchiyama’s method – associated with Furuta and Kantorovich type operator inequalities, Math

T. Furuta,Results underlogA≥logBcan be derived from ones underA≥B≥0by Uchiyama’s method – associated with Furuta and Kantorovich type operator inequalities, Math. Inequal. Appl.3(2000), no. 3, 423–436

work page 2000
[15]

Hiramatsu and Y

S. Hiramatsu and Y. Seo,Determinant for positive operators and Oppenheim’s inequality, J. Math. Inequal.15(2021), no. 4, 1637–1645

work page 2021
[16]

Hiramatsu and Y

S. Hiramatsu and Y. Seo,Determinant for positive operators and operator geometric means, Anal. Math. Phys.12(2022), Paper No. 49

work page 2022
[17]

Ligocka,Levi forms, differential forms of type(0,1)and pseudoconvexity in Banach spaces, Ann

E. Ligocka,Levi forms, differential forms of type(0,1)and pseudoconvexity in Banach spaces, Ann. Polon. Math.33(1976), 63–69

work page 1976
[18]

Mujica,Complex Analysis in Banach Spaces: Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions, North-Holland Mathematics Studies, vol

J. Mujica,Complex Analysis in Banach Spaces: Holomorphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions, North-Holland Mathematics Studies, vol. 120, North-Holland, Amsterdam, 1986

work page 1986
[19]

Peˇ cari´ c, T

J. Peˇ cari´ c, T. Furuta, J. Mi´ ci´ c Hot and Y. Seo,Mond–Peˇ cari´ c Method in Operator Inequal- ities: Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Monographs in Inequalities, vol. 1, Element, Zagreb, 2013

work page 2013
[20]

Riesz,Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel, Acta Math.48(1926), 329–343

F. Riesz,Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel, Acta Math.48(1926), 329–343

work page 1926
[21]

de la Harpe,Fuglede–Kadison determinant: theme and variations, Proc

P. de la Harpe,Fuglede–Kadison determinant: theme and variations, Proc. Natl. Acad. Sci. USA110(2013), no. 40, 15864–15877

work page 2013
[22]

Sadullaev,Plurisubharmonic measures and capacities on complex manifolds, Russian Math

A. Sadullaev,Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys36(1981), no. 4, 61–119. 32 PER ˚AHAG, RAF A LCZY˙Z, ANTTI PER ¨AL ¨A, AND JANI VIRTANEN Department of Mathematics and Mathematical Statistics, Ume˚a University, SE-901 87 Ume˚a, Sweden Email address:per.ahag@math.umu.se F aculty of Mathematics and Computer Sc...

work page 1981