Positive definite functions on complex spheres and their walks through dimensions

We provide walks through dimensions for isotropic positive definite functions defined over complex spheres. We find that the analogues of Mont\'ee and Descente operators as proposed by Beatson and zu Castell on the basis of the original Matheron operator, allow for similar walks through dimensions. We show that the Mont\'ee operators also preserve, up to a constant, the strict positive definiteness. For the Descente operators, we show that strict positive definiteness is preserved under some additional conditions, but we provide counterexamples showing that this is not true in general. We also provide a list of parametric families of (strictly) positive definite functions over complex spheres, which are important for several applications.