  The localization phenomenon for periodic unitary transition operators on a
Hilbert space consisting of square summable functions on an integer lattice
with values in a complex vector space, which is a generalization of the
discrete-time quantum walks with constant coin matrices, are discussed. It is
proved that a periodic unitary transition operator has an eigenvalue if and
only if the corresponding unitary matrix-valued function on a torus has an
eigenvalue which does not depend on the points on the torus. It is also proved
that the continuous spectrum of the periodic unitary transition operators is
absolutely continuous. As a result, it is shown that the localization happens
if and only if there exists an eigenvalue, and the long time average of the
transition probabilities coincides with the point-wise norm of the projection
of the initial state to the direct sum of eigenspaces.
