Hadamard type operators on spaces of real analytic functions in several variables

We consider multipliers on the space of real analytic functions of several variables A(Ω)A(Ω), Ω⊂RdΩ⊂Rd open, i.e., linear continuous operators for which all monomials are eigenvectors. If zero belongs to Ω these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Euler differential operators of arbitrary order are multipliers. We represent all multipliers via a kind of multiplicative convolution with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. Moreover, we represent multipliers via suitable holomorphic functions with Laurent coefficients equal to the eigenvalues of the operator. We identify in some standard cases what topology should be put on the suitable space of analytic functionals in order that the above mentioned isomorphism becomes a topological one when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on A(Ω)A(Ω). We also characterize in the same cases when the discovered topology coincides with the classical topology of bounded convergence on the space of analytic functionals. We provide several examples of multipliers and show surjectivity results for multipliers on A(Ω)A(Ω) if Ω⊂R+d.