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 THT = H, then QT symmetry is equivalent toQ-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. PACS number: 03.65.-w