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

Let A and B be bounded linear operators on a Banach space such that ABA = A 2 and BAB = B 2 .T henA and B have some spectral properties in common. This situation is studied in the present paper. 1. Terminology and motivation Throughout this paper X denotes a complex Banach space and L(X) the Ba- nach algebra of all bounded linear operators on X.F orA ∈L (X), let N (A) denote the null space of A, and let A(X) denote the range of A.W e use σ(A) ,σ p(A) ,σ ap(A) ,σ r(A) ,σ c(A )a ndρ(A) to 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 )i ssemi-Fredholm if A(X) is closed and either α(A ): = dim N (A )o rβ(A ): = codimA(X) is finite. A ∈L (X )i sFredolm if A is semi- Fredholm, α(A) < ∞ and β(A) < ∞ .T heFredholm spectrum σF (A )o fA is given by σF (A )= {λ ∈ C : λI − A is not Fredholm}. The dual space of X is denoted by X ∗ and the adjoint of A ∈L (X )b yA ∗ .