Constants related to powers of ρ-contractions

Let A be a bounded linear operator on a Hilbert space H. In this paper, we show that if A is a numerical contraction and 1≤n<∞, then ‖Ax‖=‖A²x‖=⋯=‖Aⁿx‖=2(n+1)/n for some unit vector x∈H if and only if A is unitarily similar to an operator of the form A_n⊕D, where D is a numerical contraction and [Formula presented] Moreover, we also show that if ρ>1 and A is a ρ-contraction, then lim_n⁡‖Aⁿx‖=ρ for some unit vector x∈H if and only if A is unitarily similar to an operator of the form A_ρ,∞⊕D, where D is a ρ-contraction and A_ρ,∞=[0ρ01010⋱⋱] on ℓ².