Projections in the Algebra generated by an n-Potent Operator

TL;DR

This paper characterizes all projections in the algebra generated by an n-potent operator on a Banach space, revealing their spectral structure and establishing a spectral decomposition with explicit formulas.

Contribution

It provides a complete description of projections in the algebra generated by an n-potent operator, linking them to the spectrum and establishing a spectral decomposition.

Findings

Projections form a Boolean algebra isomorphic to the power set of the spectrum minus zero.

Explicit formulas for Riesz projections are derived using resolvent expansions.

The theory is illustrated with a detailed example of 5-potent operators.

Abstract

This paper investigates the projection operators that lie in the algebra generated by powers of an $n$-potent operator $T$ on a complex Banach space, where $T^n = T$. We give a complete description of all projections in the algebra $\operatorname{comb}(T) = \text{span}\{T, T^2, \dots, T^{n-1}\}$, and prove that each such projection is uniquely determined by, and in bijection with, a subset of the nonzero spectrum of $T$. As a consequence, the family of projections in $\operatorname{comb}(T)$ forms a Boolean algebra isomorphic to the power set of $\sigma(T)\setminus\{0\}$. We also establish a spectral decomposition for $n$-potent operators in terms of their Riesz projections and derive explicit formulas for the associated Riesz projections using resolvent expansions. We give an illustration of the theory for $5$-potent operators, which highlights the algebraic and spectral structure of finite-order operators on Banach spaces.