2014/05/10 by Pal, Sourav · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1405.2436
We show an interplay between the complex geometry of the tetrablock \mathbb E and the commuting triples of operators having \mathbb E as a spectral set. We prove that every distinguished variety in the tetrablock is one-dimensional and can be represented as Ω=\ (x1,x2,x3)∈ \mathbb E : (x1,x2) ∈ σT(A1^*+x3A2 , A2^*+x3A1) \, where A1,A2 are commuting square matrices of the same order satisfying [A1^*,A1]=[A2^*,A2] and a norm condition. The converse also holds, i.e, a set of the form (\refeqn:1) is always a distinguished variety in \mathbb E. We show that for a triple of commuting operators Υ= (T1,T2,T3) having \mathbb E as a spectral set, there is a one-dimensional subvariety ΩΥ of \mathbb E depending on Υ such that von-Neumann's inequality holds, i.e, f(T1,T2,T3)≤ sup(x1,x2,x3)∈ΩΥ |f(x1,x2,x3)|, for any holomorphic polynomial f in three variables, provided that T3n→ 0 strongly as n→ ∞. The variety ΩΥ has been shown to have representation like (\refeqn:1), where A1,A2 are the unique solutions of the operator equations T1-T2^*T3=(I-T3^*T3)(1)/(2)X1(I-T3^*T3)(1)/(2) and
T2-T1^*T3=(I-T3^*T3)(1)/(2)X2(I-T3^*T3)(1)/(2). We also show that under certain condition, ΩΥ is a distinguished variety in \mathbb E. We produce an explicit dilation and a concrete functional model for such a triple (T1,T2,T3) in which the unique operators A1,A2 play the main role. Also, we describe a connection of this theory with the distinguished varieties in the bidisc and in the symmetrized bidisc.