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:

Abstract. For a bounded linear operator A on a Banach space we characterize the isolated points in the spectrum of A , the Riesz points of A , and the poles of the resolvent of A .

No abstract

Abstract. Let X be an infinite-dimensional complex Banach space and let C(X) denote the Banach algebra of all bounded linear operators on X. We write $ g (X) for the following class of operators: $g(X) = {T € C(X) : there is some S € such that TST = T and I -ST -TS is Fredholm}.Each Fredholm operator belongs to <$g(X). Operators in d> g (X) we call generalized Fredholm operators. In this paper we investigate the class

Abstract. We generalize a result of I. Vidav concerning the operator equations ABA = A 2 and BAB = B 2 .