  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 $\hat{\mathcal O}$, which is called the weak value in the real action
theory, to another expression $\langle \hat{\mathcal O}
\rangle_{\mathrm{periodic}~\mathrm{time}}$. We present two theorems stating
that $\langle \hat{\mathcal O} \rangle_{\mathrm{periodic}~\mathrm{time}}$
becomes real for $\hat{\mathcal O}$ being Hermitian with regard to a modified
inner product that makes a given non-normal Hamiltonian $\hat{H}$ 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
$\hat{H}$ 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 $B
\leq 0$ and $|B|$ is much smaller than the distances between any two real parts
of the eigenvalues of $\hat{H}$. The latter proven via a number-theoretical
argument suggests that, if our universe is periodic, then even the period could
be an adjustment parameter to be determined in the Feynman path integral. This
is a variant type of the maximization principle that we previously proposed.
