Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries

We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (${\cal P}$), time-reversal (${\cal T}$), and charge-conjugation (${\cal C}$) operators and establish the ${\cal PT}$- and ${\cal CPT}$-invariance of H.