A Nagy-Foias program for a c.n.u. $\Gamma_n$-contraction

TL;DR

This paper extends the Sz.-Nagy-Foias dilation theory to $ Gamma_n$-contractions, constructing minimal dilations, functional models, and invariants for certain classes of these operator tuples.

Contribution

It develops a Nagy-Foias type dilation and model theory specifically for $ Gamma_n$-contractions, including new invariants and conditions for dilation.

Findings

Constructed minimal $ Gamma_n$-isometric dilations.

Provided a functional model expressing $S_i$ as $C_i+PC_{n-i}$.

Established necessary and sufficient conditions for $ Gamma_n$-contraction.

Abstract

A tuple of commuting Hilbert space operators $(S_1, \dots, S_{n-1}, P)$ having the closed symmetrized polydisc \[ \Gamma_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n} z_iz_j, \cdots, \prod_{i=1}^{n}z_i\right) : |z_i|\leq 1\,, \; \; \; 1\leq i \leq n-1 \right\} \] as a spectral set is called a $\Gamma_n$-contraction. From the literature we have that a point $(s_1, \dots , s_{n-1},p)$ in $\Gamma_n$ can be represented as $s_i=c_i+pc_{n-i}$ for some $(c_1, \dots, c_{n-1}) \in \Gamma_{n-1}$. We construct a minimal $\Gamma_n$-isometric dilation for a particular class of c.n.u. $\Gamma_n$-contractions $(S_1, \cdots, S_{n-1},P)$ and obtain a functional model for them. With the help of this model we express each $S_i$ as $S_i=C_i+PC_{n-i}$, which is an operator theoretic analogue of the scalar result. We also produce an abstract model for a different class of c.n.u. $\Gamma_n$-contractions satisfying $S_i^*P=PS_i^*$ for each $i$. By exhibiting a counter example we show that such abstract model may not exist if we drop the hypothesis that $S_i^*P=PS_i^*$. We apply this abstract model to achieve a complete unitary invariant for such c.n.u. $\Gamma_n$-contractions. Additionally, we present different necessary conditions for dilation and a sufficient condition under which a commuting tuple $(S_1, \dots , S_{n-1},P)$ becomes a $\Gamma_n$-contraction. The entire program goes parallel to the operator theoretic program developed by Sz.-Nagy and Foias for a c.n.u. contraction.