On Operators all of Which Powers have the same Trace

Abstract

© 2019, Springer Science+Business Media, LLC, part of Springer Nature. We introduce the class K A,ϕ = { A∈ A: ϕ(A k ) = ϕ(A) for all k∈ ℕ} for a linear functional ϕ on an algebra A and consider the properties of this class. Also we prove the “0–1 number lemma”: if a set {zk}k=1n⊂ℂ is such that z1+…+zn=z12+…+zn2=⋯=z1n+1+…+znn+1, then z k ∈{0,1}, for all k = 1,2,…,n. This lemma helps us to show that { ϕ(A) : A∈ K A,ϕ } = { 0 , 1 , … , n} and det(A) ∈{0,1} for A= M n (ℂ) and ϕ = tr, the canonical trace. We have A = P + Z where P is a projection and Z is a nilpotent for any A∈ K A,ϕ . Assume that for a trace class operator A there exists a constant C∈ ℂ such that tr(A k ) = C for all k∈ ℕ. Then C∈ ℕ⋃ { 0 } and the spectrum σ(A) is a subset of {0,1}. Finally we give the description of all the elements of the class K A,ϕ for M 2 (ℂ).