Common Spectral Properties of Linear Operators a and B Such That Aba = a 2 and Bab = B 2

Let A and B be bounded linear operators on a Banach space such that ABA = A2 and BAB = B2. Then A 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 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 use σ(A), σp(A), σap(A), σr(A), σc(A) and ρ(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) is semi-Fredholm if A(X) is closed and either α(A) := dimN(A) or β(A) := codimA(X) is finite. A ∈ L(X) is Fredolm if A is semiFredholm, α(A) < ∞ and β(A) < ∞. The Fredholm spectrum σF (A) of A 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) by A∗. The following theorem motivates our investigation: Theorem 1.1. Let P,Q ∈ L(X) such that P 2 = P and Q = Q. If A = PQ and B = QP , then (1) ABA = A and BAB = B; (2) σ(A) {0} = σ(B) {0}; (3) σp(A) {0} = σp(B) {0}; (4) σap(A) {0} = σap(B) {0}; 2000 Mathematics Subject Classification: Primary 47A10.