Subvarieties of the tetrablock and von Neumann's inequality

Abstract

We show an interplay between the complex geometry of the tetrablock E and the commuting triples of operators having E as a spectral set. We prove that E being a 3-dimensional domain does not have any 2-dimensional distinguished variety, every distinguished variety in the tetrablock is one-dimensional and can be represented as] and a norm condition. The converse also holds, i.e, a set of the form (0.1) is always a distinguished variety in E. We show that for a triple of commuting operators Υ = (T1, T2, T3) having E as a spectral set, there is a one-dimensional subvariety ΩΥ of E depending on Υ such that von-Neumann's inequality holds, i.e, f (T1, T2, T3) ≤ supfor any holomorphic polynomial f in three variables, provided that T n 3 → 0 strongly as n → ∞. The variety ΩΥ has been shown to have representation like (0.1), where A1, A2 are the unique solutions of the operator equations. We also show that under certain condition, ΩΥ is a distinguished variety in 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.

Canonical decomposition of a tetrablock contraction and operator model

“…Complex geometry, function theory and operator theory on the tetrablock have been widely studied by a number of mathematicians [1,2,4,5,6,7,9,11,13] over past one decade because of the relevance of this domain to µ-synthesis problem and H ∞ control theory. The following result from [1] (Theorem 2.4 in [1]) characterizes points in E and E and provides a geometric description of the tetrablock.…”

The failure of rational dilation on the tetrablock

“…This domain has attracted the attention of a number of mathematicians [1,2,22,13,14,23,9,11,17] because of its relevance to µ-synthesis and H ∞ control theory. The following result from [1] (Theorem 2.4, part-( 9)) characterizes points in E and E and provides a geometric description of the tetrablock.…”