TY - JOUR

T1 - Spectral sets and distinguished varieties in the symmetrized bidisc

AU - Pal, Sourav

AU - Shalit, Orr Moshe

N1 - Funding Information:
The first author was supported in part by a postdoctoral fellowship funded in part by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. The second author is partially supported by ISF Grant No. 474/12 , by EU FP7/2007–2013 Grant No. 321749 , and by GIF Grant No. 2297-2282.6/20.1 .

PY - 2014/5/1

Y1 - 2014/5/1

N2 - 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.

AB - 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.

KW - Complete spectral set

KW - Distinguished varieties

KW - Fundamental operator

KW - Spectral set

KW - Symmetrized bidisc

KW - Von Neumann's inequality

UR - http://www.scopus.com/inward/record.url?scp=84897083259&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2013.12.022

DO - 10.1016/j.jfa.2013.12.022

M3 - Article

AN - SCOPUS:84897083259

SN - 0022-1236

VL - 266

SP - 5779

EP - 5800

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 9

ER -