  This work presents an infinite-dimensional generalization of the
correspondence between the Kullback-Leibler and R\'enyi divergences between
Gaussian measures on Euclidean space and the Alpha Log-Determinant divergences
between symmetric, positive definite matrices. Specifically, we present the
regularized Kullback-Leibler and R\'enyi divergences between covariance
operators and Gaussian measures on an infinite-dimensional Hilbert space, which
are defined using the infinite-dimensional Alpha Log-Determinant divergences
between positive definite trace class operators. We show that, as the
regularization parameter approaches zero, the regularized Kullback-Leibler and
R\'enyi divergences between two equivalent Gaussian measures on a Hilbert space
converge to the corresponding true divergences. The explicit formulas for the
divergences involved are presented in the most general Gaussian setting.
