Gaussian Exterior Dilation and Schatten Classes of Lévy Area

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 $01$, trace-class covariance suffices.