We study tempered distributions that are multipliers of the Schwartz space relative to the Moyal product. They form an algebra N under the Moyal product containing the polynomials. The elements of N are represented as infinite dimensional matrices with certain growth properties of the entries. The representation transforms the Moyal product into matrix multiplication. Each real element of N allows a resolvent map with values in tempered distributions and an associated spectral resolution. This giaes a tool to study distributions associated with symmetric, but not necessarily self-adjoint operators.