On partial isometries with circular numerical range

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 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.

Object Details

• Location: Deutsche Nationalbibliothek Frankfurt am Main
• Extent: Online-Ressource
• Language: Englisch

References and Relationships

• Bibliographic citation: On partial isometries with circular numerical range ; volume:8 ; number:1 ; year:2021 ; pages:176-186 ; extent:11 ; Concrete operators ; 8, Heft 1 (2021), 176-186 (gesamt 11)

Contributors, Places and Time

• Creator: Wegert, Elias ; Spitkovsky, Ilya

Further information

• DOI: 10.1515/conop-2020-0121
• URN: urn:nbn:de:101:1-2410301539006.314421098725
• Rights: Open Access; Der Zugriff auf das Objekt ist unbeschränkt möglich.
• Last update: 15.08.2025, 7:23 AM CEST