Properties of operators related to a $Γ_3$-contraction and unitary invariants

Sourav Pal

Published 2016-10-06Version 1

The closed symmetrized polydisc of dimension three is the set \[ \Gamma_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which $\Gamma_3$ is a spectral set is called a $\Gamma_3$-contraction. For a $\Gamma_3$-contraction $(S_1,S_2,P)$ there are two unique operators $A_1,A_2$ such that \[ S_1-S_2^*P=D_PA_1D_P\;,\; S_2-S_1^*P=D_PA_2D_P. \] The operator pair $(A_1,A_2)$ plays central role in determining the structure of a $\Gamma_3$-contraction. We shall discuss various properties of the fundamental operator pairs of $\Gamma_3$-contractions. For two operator pairs $(A_1,A_2)$ and $(B_1,B_2)$ we provide conditions under which there exists a $\Gamma_3$-contraction $(S_1,S_2,P)$ such that $(A_1,A_2)$ is the fundamental operator pair of $(S_1,S_2,P)$ and $(B_1,B_2)$ is the fundamental operator pair of its adjoint $(S_1^*,S_2^*,P^*)$. We shall show that such fundamental operator pair plays pivotal role in determining a set of unitary invariants for the $\Gamma_3$-contractions.