Structures and numerical ranges of power partial isometries

We derive a matrix model, under unitary similarity, of an n-by-n matrix A such that A,A2,…,Ak (k⩾1) are all partial isometries, which generalizes the known fact that if A is a partial isometry, then it is unitarily similar to a matrix of the form [0B0C] with B⁎B+C⁎C=I. Using this model, we show that if A has ascent k and A,A2,…,Ak−1 are partial isometries, then the numerical range W(A) of A is a circular disc centered at the origin if and only if A is unitarily similar to a direct sum of Jordan blocks whose largest size is k. As an application, this yields that, for any Sn-matrix A, W(A) (resp., W(A⊗A)) is a circular disc centered at the origin if and only if A is unitarily similar to the Jordan block Jn. Finally, examples are given to show that, for a general matrix A, the conditions that W(A) and W(A⊗A) are circular discs at 0 are independent of each other.