  The {\eta} pseudo PT symmetry theory, denoted by the symbol {\eta}, explores
the conditions under which non-Hermitian Hamiltonians can possess real spectra
despite the violation of PT symmetry, that is the adjoint of H, denoted
H^{{\dag}} is expressed as H^{{\dag}}=PTHPT. This theory introduces a new
symmetry operator, {\eta}=PT{\eta}, which acts on the Hilbert space. The {\eta}
pseudo PT symmetry condition requires the Hamiltonian to commute with the
{\eta} operator, leading to real eigenvalues. We discuss some general
implications of our results for the coupled non hermitian harmonic oscillator.
