Invariant kernels on the space of complex covariance matrices

The present work sets forth the analytical tools which make it possible to construct and compute invariant kernels on the space of complex covariance matrices. The main result is the $\mathrm{L}^1$--\,Godement theorem, which is used to obtain a general analytical expression that yields any invariant kernel which is, in a certain natural sense, also integrable. Using this expression, one can design and explore new families of invariant kernels, all while incurring a rather moderate computational effort. The expression comes in the form of a determinant (it is a determinantal expression), and is derived from the notion of spherical transform, which arises when the space of complex covariance matrices is considered as a Riemannian symmetric space.