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arXiv · 2109.08481

On partial isometries with circular numerical range

Elias Wegert·Ilya M. Spitkovsky
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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.

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Elias Wegert, Ilya M. Spitkovsky. 2021-09-17. On partial isometries with circular numerical range. https://arxiv.org/abs/2109.08481

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