Spectral sets and distinguished varieties in the symmetrized bidisc

Sourav Pal, Orr Moshe Shalit

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


We show that for every pair of matrices (S, P), having the closed symmetrized bidisc Γ as a spectral set, there is a one dimensional complex algebraic variety Λ in Γ such that for every matrix valued polynomial f(z1, z2), {norm of matrix}f(S,P){norm of matrix}≤max(z1,z2)∈Λ{norm of matrix}f(z1,z2){norm of matrix}. The variety Λ is shown to have the determinantal representation Λ={(s,p)∈Γ: det(F+pF*-sI)=0}, where F is the unique matrix of numerical radius not greater than 1 that satisfies S-S*P=(I-P*P)1/2F(I-P*P)1/2. When (S, P) is a strict Γ-contraction, then Λ is a distinguished variety in the symmetrized bidisc, i.e. a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.

Original languageEnglish
Pages (from-to)5779-5800
Number of pages22
JournalJournal of Functional Analysis
Issue number9
StatePublished - 1 May 2014


  • Complete spectral set
  • Distinguished varieties
  • Fundamental operator
  • Spectral set
  • Symmetrized bidisc
  • Von Neumann's inequality

ASJC Scopus subject areas

  • Analysis


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