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Abstract

Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence ${x^{n}(x-1)}_{n ∈ ℕ}$ for an arbitrary x ∈ A, using σ(x) and resolvent conditions. Under these circumstances, we investigate elements in the peripheral spectrum, and give further conclusions, also involving the behaviour of ${x^{n}}_{n ∈ ℕ}$ and ${1/n ∑_{k=0}^{n-1} x^{k}}_{n ∈ ℕ}$.