A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. Quasi-Hermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a ${ \rm su}(1,1/1) \sim { \rm osp}(2/2, \R)$ superalgebra. This allows the construction of new non-Hermitian Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.
Categories: math-ph hep-th math.MP quant-ph