On partial isometries with circular numerical range

2021/09/17 by Wegert, Elias, Spitkovsky, Ilya M.
#30J10 #33E05 #47A12 #47A20 #51N15 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2109.08481

Abstract

In their LAMA'2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂⁿ cannot have a circular numerical range with a non-zero center, and proved this conjecture for n≤ 4. We prove it for operators with 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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