Power Partial Isometry Index and Ascent of a Finite Matrix
Abstract
We give a complete characterization of nonnegative integers $j$ and $k$ and a positive integer $n$ for which there is an $n$by$n$ matrix with its power partial isometry index equal to $j$ and its ascent equal to $k$. Recall that the power partial isometry index $p(A)$ of a matrix $A$ is the supremum, possibly infinity, of nonnegative integers $j$ such that $I, A, A^2, \ldots, A^j$ are all partial isometries while the ascent $a(A)$ of $A$ is the smallest integer $k\ge 0$ for which $\ker A^k$ equals $\ker A^{k+1}$. It was known before that, for any matrix $A$, either $p(A)\le\min\{a(A), n1\}$ or $p(A)=\infty$. In this paper, we prove more precisely that there is an $n$by$n$ matrix $A$ such that $p(A)=j$ and $a(A)=k$ if and only if one of the following conditions holds: (a) $j=k\le n1$, (b) $j\le k1$ and $j+k\le n1$, and (c) $j\le k2$ and $j+k=n$. This answers a question we asked in a previous paper.
 Publication:

arXiv eprints
 Pub Date:
 November 2013
 arXiv:
 arXiv:1311.2361
 Bibcode:
 2013arXiv1311.2361G
 Keywords:

 Mathematics  Functional Analysis;
 15A18;
 15B99
 EPrint:
 11 pages