THE SPHERICAL MEAN-VALUE OPERATOR FOR COMPACT SYMMETRICAL SPACES

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 eigenva lues lambda for L(r)f = lambda f are explicitly determined in terms of the elementary spherical functions associated with the symmetric spac e. Alternative proofs are also provided for some results of T. Sunada regarding the special eigenvalues +1 and -1 using a purely harmonic an alytic point of view.