Projections in the Algebra generated by an n-Potent Operator

2025/12/27 by Monika, Dey, Priyadarshi, Easley, Zachary
#46 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2512.22497

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 Tⁿ = T. We give a complete description of all projections in the algebra comb(T) = span\T, T², …, 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 comb(T) forms a Boolean algebra isomorphic to the power set of σ(T)∖\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.

Citations

• Spectral decomposition of power-bounded operators: The finite spectrum case

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