Hadamard multipliers on spaces of real analytic functions

We consider multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We characterize sequences of complex numbers which are sequences of eigenvalues for some multiplier. We characterize invertible multipliers, in particular, we find which Euler differential operators of infinite order have global analytic solutions on the real line. We present a number of examples where our theory applies. In some cases we give algorithms for solving the respective equations. Perturbation results for solvability are presented.