LAUSR

Abstract Let K 2 = [ 0 2 0 0 ] , K n be the n × n weighted shift matrix with weights 2 , 1 , … , 1 ︸ n − 3 , 2 for all n ≥ 3 , and K ∞ be the weighted shift operator with weights 2 , 1 , 1 , 1 , … . In this paper, we show that if an n × n nonzero matrix A satisfies W ( A k ) = W ( A ) for all 1 ≤ k ≤ n , then W ( A ) cannot be a (nondegenerate) circular disc. Moreover, we also show that W ( A ) = W ( A n − 1 ) = { z ∈ C : | z | ≤ 1 } if and only if A is unitarily similar to K n . Finally, we prove that if T is a numerical contraction on an infinite-dimensional Hilbert space H, then lim n → ∞ ⁡ ‖ T n x ‖ = 2 for some unit vector x ∈ H if and only if T is unitarily similar to an operator of the form K ∞ ⊕ T ′ with w ( T ′ ) ≤ 1 .