Tripotents in algebras: Invertibility and hyponormality

Abstract

Let A be a unital algebra over complex field ℂ, I be the unit of A. An element A ∈ A is called tripotent if A ³ = A. Let A ^tri = {A ∈ A: A ³ = A}. We show that A ∈ A ^tri if and only if I ± A − A ² ∈ A ^tri. We study invertibility properties of elements I + λA with A ∈ A ^tri and λ ∈ ℂ \ {−1,1}. Let X be a Banach space with the approximation property and A, B ∈ B(X)^tri. If A − B is a nuclear operator then tr(A − B) ∈ ℂ. We show that if H is a Hilbert space and an operator A ∈ B(H)^tri is hyponormal or cohyponormal then A = A*. We also prove that every A ∈ B(H)^tri similar to a Hermitian tripotent.