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.
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