Quantum mechanics using two auxiliary inner products.

The current applications of non-Hermitian but ${\cal PT}-$symmetric Hamiltonians $H$ cover several, mutually not too closely connected subdomains of quantum physics. Mathematically, the split between the open and closed systems can be characterized by the respective triviality and non-triviality of an auxiliary inner-product metric $\Theta=\Theta(H)$. With our attention restricted to the latter, mathematically more interesting unitary-evolution case we show that the intuitive but technically decisive simplification of the theory achieved via an "additional" ${\cal PCT}-$symmetry constraint upon $H$ can be given a deeper mathematical meaning via introduction of a certain second auxiliary inner product.