2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159617Among ${\cal P}$-pseudo-Hermitian Hamiltonians $H ={\cal P}^{-1} H^\dagger \cal P}$ with real spectra, the ''weakly pseudo-Hermitian" ones (i.e., those employing non-self-adjoint ${\cal P} \neq {\cal P}^\dagger$) form a remarkable subfamily. We list some reasons why it deserves a special attention. In particular we show that whenever ${\cal P} \neq {\cal P}^\dagger$, the current involutive operator of charge ${\cal C}$ gets complemented by a nonequivalent alternative involutive quasiparity operator ${\cal Q}$. We show how, in this language, the standard quantum mechanics is restored via the two alternative inner products in the physical Hilbert space of states, with $<ψ_1|{\cal PQ}|ψ_2>=< ψ_1|{\cal CP}|ψ_2>$.This is a contribution to the Proc. of the 3-rd Microconference "Analytic and Algebraic Methods III"(June 19, 2007, Prague, Czech Republic), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/Mathematical PhysicsQuantum Physics46C20; 47B50; 81P10; 81R05text