Eigenfunctions of expected value operators in the Wishart distribution, II

Let V = (vij) denote the k × k symmetric scatter matrix following the Wishart distribution W(k, n, Σ). The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar-valued functions f(V) such that (Enf)(V) = λn,kf(V). A finite sequence of polynomial eigenspaces, EP spaces, exists whose direct sum is the space of all homogeneous polynomials. These EP subspaces are invariant and irreducible under the action of the congruence transformation V → T′VT. Each of these EP subspaces contains an orthogonally invariant subspace of dimension one. The number of EP subspaces is determined and eigenvalues are computed. Bi-linear expansions of |I + VA|−n2 and (tr VA)r into eigenfunctions are given. When f(V) is an EP polynomial, then f(V−1) is an EP function. These EP subspaces are identical to the more abstractly defined polynomial subspaces studied by James.