Numerical range, Blaschke products and Poncelet polygons

Elias Wegert (Technische Universität Bergakademie Freiberg)

14-Jun-2022, 13:00-14:00 (21 months ago)

Abstract: (Joint work with Ilya Spitkovsky, New York University Abu Dhabi)

In 2016, Gau, Wang and Wu conjectured that a partial isometry A acting on a $n$-dimensional complex Hilbert space cannot have a circular numerical range with a non-zero center. In this talk we present a proof for operators with rank $A=n-1$ and any n. It is based on the unitary similarity of A to a compressed shift operator generated by a finite Blaschke product $B(z)$. We then use the description of the numerical range by Poncelet polygons associated with $zB(z)$, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenters of the vertices of Poncelet polygons involving elliptic functions.

complex variablesdynamical systemsnumerical analysis

Audience: researchers in the topic


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