2021/05/27 by S. H. Kulkarni, Kulkarni, S. H.
Mathematics · #46B99 #47A05 #Advanced Banach Space Theory #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2105.12959
openalex publication_date 2021/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a complex unital Banach algebra with unit 1. An element a∈ A is said to be of \textitG1-class if ‖(z-a)-1‖=(1)/(d(z,σ(a))) ∀ z∈ ℂ∖ σ(a). Here d(z, σ(a)) denotes the distance between z and the spectrum σ(a) of a. Some examples of such elements are given and also some properties are proved. It is shown that a G1-class element is a scalar multiple of the unit 1 if and only if its spectrum is a singleton set consisting of that scalar. It is proved that if T is a G1 class operator on a Banach space X, then every isolated point of σ(T) is an eigenvalue of T. If, in addition, σ(T) is finite, then X is a direct sum of eigenspaces of T.