QT-symmetry and weak pseudo-hermiticity

For an invertible (bounded) linear operator Q acting in a Hilbert space H, we consider the consequences of the QT-symmetry of a non-hermitian Hamiltonian H: H→H where T is the time-reversal operator. If H is symmetric in the sense that TH†T=H, then QT-symmetry is equivalent to Q-1-weak-pseudo-hermiticity. But in general this equivalence does not hold. We show this using some specific examples. Among these is a large class of non-PT-symmetric Hamiltonians that share the spectral properties of PT-symmetric Hamiltonians