On low-dimensional partial isometries

TL;DR

This paper investigates properties of n-by-n partial isometries, showing that certain geometric and algebraic features hold for small sizes but fail for larger matrices.

Contribution

It establishes that partial isometries are generic for unitarily irreducible cases and that nilpotent partial isometries have circular numerical ranges for small sizes, with these properties failing at larger sizes.

Findings

Properties hold for n ≤ 4 but fail for n ≥ 5.

Nilpotent partial isometries have circular numerical ranges for small n.

Unitarily irreducible partial isometries are generic for small n.

Abstract

Two statements concerning $n$-by-$n$ partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for $n\leq 4$ but fail starting with $n=5$.