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On the differences of the consecutive powers of Banach algebra elements

Helmuth Rönnefarth — 1997

Banach Center Publications

Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence ${x^{n} (x - 1)}_{n \in ℕ}$ 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 ${x^{n}}_{n \in ℕ}$ and ${1 / n \sum_{k = 0}^{n - 1} x^{k}}_{n \in ℕ}$ .

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