Periodic Points of Diagonal and Permutation Operators

TL;DR

This work analyzes periodic points $P(T)$ for bounded operators on infinite-dimensional separable complex Hilbert spaces, with a focus on normal, diagonal, and permutation operators. Using spectral calculus and eigenspace decompositions, it shows that a normal operator has no nonzero periodic points precisely when its spectrum avoids the root-of-unity set $G$, and that $P(T)=H$ forces a finite-exponent, diagonalizable, unitary operator. It then provides a complete description of $P(T)$ for diagonal and for permutation operators, including criteria for when $P(T)$ is closed, equals the whole space, or is dense, and establishes density results for diagonal unitary operators with $P(T)=H$ among diagonal unitaries. These results connect spectral properties to dynamical behavior and supply constructive tools for shaping periodic dynamics in Hilbert-space operator theory.

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.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Example 2.4