Structures and numerical ranges of power partial isometries

Abstract

We derive a matrix model, under unitary similarity, of an n-by-n matrix A such that A, A(2),, A(k) (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 [(0 B)(0 C)] with B*B +C*C =1. Using this model, we show that if A has ascent k and A, A(2),..., 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 S-n-matrix A, W (A) (resp., W (A circle times A)) is a circular disc centered at the origin if and only if A is unitarily similar to the Jordan block J(n). Finally, examples are given to show that, for a general matrix A, the conditions that W (A) and W (A circle times A) are circular discs at 0 are independent of each other. (C) 2013 Elsevier Inc. All rights reserved.