Unitary part of a contraction

For a contraction A on a Hilbert space H, we define the index j ( A ) (resp., k ( A ) ) as the smallest nonnegative integer j (resp., k) such that ker ( I − A j ∗ A j ) (resp., ker ( I − A k * A k ) ∩ ker ( I − A k A k ∗ ) ) equals the subspace of H on which the unitary part of A acts. We show that if n = dim H < ∞ , then j ( A ) ⩽ n (resp., k ( A ) ⩽ ⌈ n / 2 ⌉ ), and the equality holds if and only if A is of class S n (resp., one of the three conditions is true: (1) A is of class S n , (2) n is even and A is completely nonunitary with ‖ A n − 2 ‖ = 1 and ‖ A n − 1 ‖ < 1 , and (3) n is even and A = U ⊕ A ′ , where U is unitary on a one-dimensional space and A ′ is of class S n − 1 ).