Rigged Hilbert space approach for non-Hermitian systems with positive definite metric

We investigate Dirac's bra-ket formalism based on a rigged Hilbert space for non-Hermite quantum system with positive-definite metric. First, the space, characterized by metric, is established. With aid of nuclear spectral theorem obtained expansions are shown bra-kets generalized eigenvectors quasi-Hermite operator. The utilized to endow complete bi-orthogonal and transformation theory between Hermite systems. As an example application, we show specific description our treatment some parity-time symmetrical