IUMJ

Title: The tetrablock as a spectral set

Authors: Tirthankar Bhattacharyya

Issue: Volume 63 (2014), Issue 6, 1601-1629

Abstract:

The tetrablock, roughly speaking, is the set of all linear fractional maps that map the open unit disc to itself. A formal definition of this inhomogeneous domain is given below. This paper considers triples of commuting bounded operators $(A,B,P)$ that have the tetrablock as a spectral set. Such a triple is named a tetrablock contraction. The motivation comes from the success of model theory in another inhomogeneous domain, namely, the symmetrized bidisc $\Gamma$. A pair of commuting bounded operators $(S,P)$ with $\Gamma$ as a spectral set is called a \emph{$\Gamma$-contraction}, and always has a dilation. The two domains are related intricately as the Lemma 3.2 below shows. Given a triple $(A,B,P)$ as above, we associate with it a pair $(F_1,F_2)$, called its \emph{fundamental operators}. We show that $(A,B,P)$ dilates if the fundamental operators $F_1$ and $F_2$ satisfy certain commutativity conditions. Moreover, the dilation space is no bigger than the minimal isometric dilation space of the contraction $P$. Whether these commutativity conditions are necessary, too, is not known. what we have shown is that if there is a tetrablock isometric dilation on the minimal isometric dilation space of $P$, then those commutativity conditions necessarily get imposed on the fundamental operators. En route, we decipher the structure of a tetrablock unitary (this is the candidate as the dilation triple) and a tertrablock isometry (the restriction of a tetrablock unitary to a joint invariant subspace). We derive new results about $\Gamma$-contractions and apply them to tetrablock contractions. The methods applied are motivated by [T. Bhattacharyya, S. Pal, and S. Shyam Roy, \emph{Dilations of \Gamma-contractions by solving operator equations}, Adv. Math. \textbf{230} (2012), no. 2, 577--606]. Although the calculations are lengthy and more complicated, they beautifully reveal that the dilation depends on the mutual relationship of the two fundamental operators, so that certain conditions need to be satisfied. The question of whether all tetrablock contractions dilate or not is unresolved.