  In this work we present a general formalism to treat non-Hermitian and
noncommutative Hamiltonians. This is done employing the phase-space formalism
of quantum mechanics, which allows to write a set of robust maps connecting the
Hamitonians and the associated Wigner functions to the different Hilbert space
structures, namely, those describing the non-Hermitian and noncommutative,
Hermitian and noncommutative, and Hermitian and commutative systems. A general
recipe is provided to obtain the expected values of the more general
Hamiltonian. Finally, we apply our method to the harmonic oscillator under
linear amplification and discuss the implications of both non-Hermitian and
noncommutative effects.
