  We use the theory of pseudo-Hermitian operators to address the problem of the
construction and classification of positive-definite invariant inner-products
on the space of solutions of a Klein-Gordon type evolution equation. This
involves dealing with the peculiarities of formulating a unitary quantum
dynamics in a Hilbert space with a time-dependent inner product. We apply our
general results to obtain possible Hilbert space structures on the solution
space of the equation of motion for a classical simple harmonic oscillator, a
free Klein-Gordon equation, and the Wheeler-DeWitt equation for the
FRW-massive-real-scalar-field models.
