On partial isometries with circular numerical range

Elias Wegert, Ilya Spitkovsky

Research output: Contribution to journalArticlepeer-review


In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on n 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 SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB 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.

Original languageEnglish (US)
JournalConcrete Operators
Issue number1
StatePublished - Jan 1 2021


  • Blaschke products
  • Poncelet's porism
  • elliptic functions
  • interpolation
  • numerical range
  • partial isometry

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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