  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.
