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Emphasizing the physical constraints on the formulation of a quantum theorybased on the standard measurement axiom and the Schroedinger equation, wecomment on some conceptual issues arising in the formulation of PT-symmetricquantum mechanics. In particular, we elaborate on the requirements of theboundedness of the metric operator and the diagonalizability of theHamiltonian. We also provide an accessible account of a Krein-space derivationof the CPT-inner product that was widely known to mathematicians since 1950's.We show how this derivation is linked with the pseudo-Hermitian formulation ofPT-symmetric quantum mechanics.Comment: published version, 17 page

Krein-space formulation of $$\mathcal{P}\mathcal{T}$$ symmetry, $$\mathcal{C}\mathcal{P}\mathcal{T}$$ -inner products, and pseudo-Hermiticity