Canonical decomposition of a tetrablock contraction and operator model

Abstract

Abstract. A triple of commuting operators for which the closed tetrablock E is a spectral set is called a tetrablock contraction or an Econtraction. The set E is defined as E = {(x1, x2, x3) ∈ C 3 : 1−zx1−wx2+zwx3 = 0 whenever |z| ≤ 1, |w| ≤ 1}.We show that every E-contraction can be uniquely written as a direct sum of an E-unitary and a completely non-unitary E-contraction. It is analogous to the canonical decomposition of a contraction operator into a unitary and a completely non-unitary contraction. We produce a concrete operator model for such a triple satisfying some conditions.

A generalized Schwarz lemma for two domains related to μ-synthesis

“…We mention here that the domain tetrablock was introduced by these three mathematicians in [1] and that this domain has deep connection with the most appealing and di cult problem of µ-synthesis (see [5] for further details). Since then the tetrablock has attracted considerable attention [5,6,[10][11][12][15][16][17]. So, our rst main result is the following Schwarz lemma for the tetrablock.…”