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

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=A2 and BAB=B2. Then A and B have some spectral properties in common. This situation is studied in the present paper.