On Operators all of Which Powers have the same Trace

We introduce the class $K_{\mathcal {A}, \phi }=\{A \in \mathcal {A}: \phi (A^{k})=\phi (A)$ for all $k \in \mathbb {N}\}$ for a linear functional ϕ on an algebra $\mathcal {A}$ and consider the properties of this class. Also we prove the “0–1 number lemma”: if a set $\{z_{k}\}_{k = 1}^{n} \subset \mathbb {C}$ is such that $z_{1}+\ldots +z_{n}={z_{1}^{2}}+\ldots +{z_{n}^{2}}=\cdots =z_{1}^{n + 1}+\ldots +z_{n}^{n + 1}$ , then zk ∈{0,1}, for all k = 1,2,…,n. This lemma helps us to show that $\{\phi (A): A \in K_{\mathcal {A}, \phi }\}=\{0, 1, \ldots , n\}$ and det(A) ∈{0,1} for $\mathcal {A}=\mathbb {M}_{n}(\mathbb {C})$ and ϕ = tr, the canonical trace. We have A = P + Z where P is a projection and Z is a nilpotent for any $A \in K_{\mathcal {A}, \phi }$ . Assume that for a trace class operator A there exists a constant $C \in \mathbb {C}$ such that tr(Ak) = C for all $k \in \mathbb {N}$ . Then $C \in \mathbb {N}\bigcup \{0\}$ and the spectrum σ(A) is a subset of {0,1}. Finally we give the description of all the elements of the class $ K_{\mathcal {A}, \phi }$ for $\mathbb {M}_{2}(\mathbb {C})$ .