Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces

Let $H$ be a complex Hilbert space and let $\mathrm{\Omega }\subset H$ 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 ${\mathrm{\Delta }}_x(A):=\exp (⟨(\log A)x,x⟩)$, and we extend this naturally to non-invertible positive operators. We show that, for strictly positive operators, inequalities for ${\mathrm{\Delta }}_x$ precisely describe the chaotic order $\log A\ge \log B$, and we combine this observation with Kantorovich–Specht type bounds for positive operators. For $u\in \mathcal{P}\mathcal{S}\mathcal{H}(\mathrm{\Omega })\cap C^2(\mathrm{\Omega })$ we define the \emph{Fujii–Seo determinant density} $$\mathrm{FSD}(u)(a):=\underset{\mathrm{\parallel }x\mathrm{\parallel }=1}{\inf }{\mathrm{\Delta }}_x\text{\hspace{0.17em}⁣}(D^{\mathrm{\prime }}{D}^{\mathrm{\prime }\mathrm{\prime }}u(a)),a\in \mathrm{\Omega },$$ and identify it with the lower spectral endpoint $\inf \sigma (D^{\mathrm{\prime }}{D}^{\mathrm{\prime }\mathrm{\prime }}u(a))$. Thus, $\mathrm{FSD}(u)$ 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 $\mathrm{FSD}(u)\equiv 0$, 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.