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We provide walks through dimensions for isotropic positive definite functionsdefined over complex spheres. We show that the analogues of Mont\'ee andDescente operators as proposed by Beatson and zu Castell [J. Approx. Theory 221(2017), 22-37] on the basis of the original Matheron operator [Les variablesr\'egionalis\'ees et leur estimation, Masson, Paris, 1965], allow for similarwalks through dimensions. We show that the Mont\'ee operators also preserve, upto a constant, strict positive definiteness. For the Descente operators, weshow that strict positive definiteness is preserved under some additionalconditions, but we provide counterexamples showing that this is not true ingeneral. We also provide a list of parametric families of (strictly) positivedefinite functions over complex spheres, which are important for severalapplications.

Positive Definite Functions on Complex Spheres and their Walks through Dimensions