When M is a compact symmetric space, the spherical mean value operator L r (for a fixed r > 0) acting on L 2 (M) is considered. The eigenvalues λ for L r f = λ/ are explicitly determined in terms of the elementary spherical functions associated with the symmetric space. Alternative proofs are also provided for some results of T. Sunada regarding the special eigenvalues -hi and -1 using a purely harmonic analytic point of view.