Abstract
Let T be an operator on a separable Hubert space. We show that if T is a compact operator and satisfies ∣T n ∣ = ∣T∣ n for some n, then T is normal, and that if T is a bounded operator and satisfies ∣T n ∣ = ∣T∣ n for n =i,i + 1, k,k + 1 (1 ≤ i <k), then the polar decomposition of T is commutative. For a closed operator we obtain the analogous results.
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