On partial isometries with circular numerical range

TL;DR

This paper proves that partial isometries with rank n-1 in complex n-dimensional space cannot have a circular numerical range with a non-zero center, extending previous results and using advanced geometric and functional analysis tools.

Contribution

It establishes the conjecture for operators of rank n-1 in any dimension, using unitary similarity to a compressed shift operator and geometric descriptions of numerical ranges.

Findings

Proved the conjecture for rank n-1 operators in all dimensions.

Connected numerical range geometry to Poncelet polygons and Blaschke products.

Derived explicit formulas involving elliptic functions for the barycenter of Poncelet polygon vertices.

Abstract

In their LAMA'2016 paper Gau, Wang and Wu conjectured that a partial isometry $A$ acting on $\mathbb{C}^n$ cannot have a circular numerical range with a non-zero center, and proved this conjecture for $n\leq 4$. We prove it for operators with $\mathrm{rank}\,A=n-1$ and any $n$. The proof is based on the unitary similarity of $A$ to a compressed shift operator $S_B$ generated by a finite Blaschke product $B$. We then use the description of the numerical range of $S_B$ as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.