Periodic Points of Diagonal and Permutation Operators

TL;DR

This paper characterizes normal operators with specific periodic point properties on Hilbert spaces, analyzes the structure of periodic points for diagonal and permutation operators, and shows density results among diagonal operators.

Contribution

It provides new conditions for the absence or presence of nonzero periodic points in normal operators and explores the structure and density of such operators in the space of diagonal operators.

Findings

Normal operators with no nonzero periodic points are characterized.

The structure of periodic points for diagonal and permutation operators is described.

Diagonal operators with all space as periodic points are dense among unitary diagonal operators.

Abstract

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.