Periodic Points of Diagonal and Permutation Operators

We first give a condition for a normal operator on a Hilbert space to have no nonzero periodic points, then we give a characterization of normal operators with the whole space as periodic points. We proceed to study the structure of periodic points of the diagonal operators and the permutation operators with examples. Moreover, it is also shown that the set of all diagonal operators with the whole space as periodic points is dense in the set of all unitary diagonal operators.