# Power Partial Isometry Index and Ascent of a Finite Matrix

Research paper by **Hwa-Long Gau, Pei Yuan Wu**

Indexed on: **11 Nov '13**Published on: **11 Nov '13**Published in: **Mathematics - Functional Analysis**

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#### 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 $p(A)\le\min\{a(A), n-1\}$ 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) $j=k\le n-1$, (b) $j\le k-1$ and $j+k\le n-1$,
and (c) $j\le k-2$ and $j+k=n$. This answers a question we asked in a previous
paper.