Reality from maximizing overlap in the periodic complex action theory

We study the periodic complex action theory (CAT) by imposing a periodic condition in the future-included CAT where the time integration is performed from the past to the future, and extend a normalized matrix element of an operator (O) over cap, which is called the weak value in the real action theory, to another expression (periodic time). We present two theorems stating that (periodic time) becomes real for (O) over cap being Hermitian with regard to a modified inner product that makes a given non-normal Hamiltonian (H) over cap normal. The first theorem holds for a given period t(p) in a case where the number of eigenstates having the maximal imaginary part B of the eigenvalues of (H) over cap is just one, while the second one stands for t(p) selected such that the absolute value of the transition amplitude is maximized in a case where B