A Nagy–Foias Program for a C.N.U. $$\Gamma _n$$ -Contraction

Abstract

A tuple of commuting Hilbert space operators \((S_1, \ldots , S_{n-1}, P)\) having the closed symmetrized polydisc

$$\begin{aligned} \Gamma _n = \left\{ \left( \sum _{i=1}^{n}z_i, \sum \limits _{1\le i<j\le n} z_iz_j, \ldots , \prod _{i=1}^{n}z_i\right) : |z_i|\le 1, \; \; \; 1\le i \le n-1 \right\} \end{aligned}$$

as a spectral set is called a \(\Gamma _n\)-contraction. From the literature we have that a point \((s_1, \ldots , s_{n-1},p)\) in \(\Gamma _n\) can be represented as \(s_i=c_i+pc_{n-i}\) for some \((c_1, \ldots , c_{n-1}) \in \Gamma _{n-1}\). We construct a minimal \(\Gamma _n\)-isometric dilation for a particular class of c.n.u. \(\Gamma _n\)-contractions \((S_1, \ldots , S_{n-1},P)\) 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 \((S_1, \ldots , S_{n-1},P)\) 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.