  PT-symmetric quantum mechanics is an alternative formulation of quantum
mechanics in which the mathematical axiom of Hermiticity (transpose and complex
conjugate) is replaced by the physically transparent condition of space-time
reflection symmetry (PT-symmetry). A Hamiltonian H is said to be PT-symmetric
if it commutes with the operator PT. The key point of PT-symmetric quantum
theory is to build a new positive definite inner product on the given Hilbert
space so that the given Hamiltonian is Hermitian with respect to the new inner
product.
  The aim of this note is to give further mathematical discussions on this
theory. Especially, concepts of PT-frames, CPT-frames on a Hilbert space and
for a Hamiltonian are proposed, their existence and constructions are
discussed.
