  A class of integral transforms, on the planar Gaussian Hilbert space with
range in the weighted Bergman space on the bi-disk, is defined as the dual
transforms of the 2d fractional Fourier transform associated with the Mehler
function for It\^o--Hermite polynomials. Some spectral properties of these
transforms are investigated. Namely, we study their boundedness and identify
their null spaces as well as their ranges. Such identification depends on the
zeros set of It\^o--Hermite polynomials. Moreover, the explicit expressions of
their singular values are given and compactness and membership in p-Schatten
class are studied. The relationship to specific fractional Hankel transforms is
also established
