We study the skew-adjoint Hilbert--Schmidt operator obtained by evaluating a fixed Hilbert two-form on a centered Gaussian linear map whose covariance is the inner product on its domain multiplied by a fixed nonnegative trace-class operator. We compute the expected regularized determinant of the resulting random operator and compare the random determinant with this expectation by dilation of the scalar parameter. For each fixed covariance and two-form, a single dilation almost surely applies to every nonnegative parameter value and yields simultaneous bounds for all Schatten norms and quasi-norms of orders $0<p<2$. The least admissible dilation satisfies a universal weak-$L^1$ estimate with optimal tail exponent. For Wiener area, a rank-one perturbation identity identifies the mean determinant with the regularized determinant of an explicit deterministic skew-adjoint operator and reduces Schatten membership to conditions on the covariance eigenvalues. Consequently, for each $0<p<1$, the area belongs almost surely to the Schatten class of order $p$ if and only if the covariance of the Wiener process does. At $p=1$, the criterion is finite covariance entropy; for $p>1$, trace-class covariance suffices.
This paper studies continuous covariant kernel families over hulls of Delone sets of finite local complexity using graded Wiener-algebra methods. The norm given by the least common weighted summable off-diagonal profile is identified isometrically with a graded $\ell^1$-norm. The resulting Banach $*$-algebras are symme...
Does the Neumann--Poincar\'e spectrum determine a planar domain up to an extended M\"obius transformation? We construct a pair showing that the answer is no, even for analytic, strictly convex, algebraic boundaries. Their operators are similar. We then prove a positive result for prescribed finite-dimensional families...
We study generalized Volterra companion operators on Bergman spaces over smoothly bounded convex domains of finite type. A derivative Carleson embedding gives boundedness and compactness criteria between reflexive Bergman spaces. When the target exponent is smaller than the source exponent, boundedness already implies...
We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact K\"ahler manifolds that provides the link between the $L^\infty$-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analy...
We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}...
We prove strict P\'olya inequalities for every positive Neumann eigenvalue of every Euclidean annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. Explicit polynomial positivity arguments and angular-l...
Guo-Wei Dai, Ying-Xin Sun· 1 citation
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