A Nagy-Foias program for a c.n.u. ${\Gamma }_n$-contraction

Abstract

A tuple of commuting Hilbert space operators $(S_1,\dots ,S_{n-1},P)$ having the closed symmetrized polydisc \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n} z_iz_j, \cdots, \prod_{i=1}^{n}z_i\right) : |z_i|\leq 1\,, \; \; \; 1\leq i \leq n-1 \right\} \] 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 $s_i=c_i+p{c}_{n-i}$ 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 $S_i=C_i+P{C}_{n-i}$, 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 $S_i^{\ast }P=P{S}_i^{\ast }$ for each $i$. By exhibiting a counter example we show that such abstract model may not exist if we drop the hypothesis that $S_i^{\ast }P=P{S}_i^{\ast }$. 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.

22 Pages, Submitted to Journal

References in corpus (1)

• Operator theory on the tetrablock