The spherical mean value operator for compact symmetric spaces

Abstract

When M is a compact symmetric space, the spherical mean value operator L_r(for a fixed r > 0) acting on L²(M) is considered. The eigenvalues λ for L_rf = λf 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 +1 and −1 using a purely harmonic analytic point of view.