Common spectral properties of linear operators a and b such that ABA=A² and BAB=B²

Abstract

Abstract. Let A and B be bounded linear operators on a Banach space such that ABA = A 2 and BAB = B 2 . Then A and B have some spectral properties in common. This situation is studied in the present paper. Terminology and motivationThroughout this paper X denotes a complex Banach space and L(X) the Banach algebra of all bounded linear operators on X. For A ∈ L(X), let N (A) denote the null space of A, and let A(X) denote the range of A. We useto denote spectrum, the point spectrum, the approximate point spectrum, the residual spectrum, the continuous spectrum and the resolvent set of A, respectively. An operator A ∈ L(X) is semi-Fredholm if A(X) is closed and either α(A)The dual space of X is denoted by X * and the adjoint of A ∈ L(X) by A * . The following theorem motivates our investigation:

On the Dunford Property (C) for Bounded Linear Operators RS and SR

“…In this case R, S, SR and RS share many spectral properties [9,10], and local spectral properties as decomposability, property (β) and SVEP [6]. In this section we consider the permanence of property (C) in this context.…”

A note on the common spectral properties for bounded linear operators

“…In the following two examples, the common spectral properties for AC and BA can only followed directly from the above results, but not from the corresponding ones in [7,9,15,16,19].…”

On subscalarity of some 2×2 class A operator matrices

“…Let D be any open disk in C containing σ (A) = σ (B) (see [19]). By [6], there exist a constant C > 0 and a positive integer k such that …”

“…Let D be any open disk in C containing σ (AB) = σ (A) (see [19]). Let h n ∈ H ⊕ H and f n ∈ W k+2 (D, H⊕H) be sequences such that lim n→∞ (AB − z)f n + 1 ⊗ h n W k+2 (D,H⊕H) = 0. for i = 1, 2.…”