Uniqueness of fundamental operators for Theta_n-contractions

Determine whether the tuples of fundamental operators associated with a Theta_n-contraction are uniquely determined by the operator tuple. Specifically, establish whether, for a Theta_n-contraction mathbf T=(T_1,ldots,T_n), the tuples (A_0^{(i)},ldots,A_p^{(i)}) in the fundamental equations are uniquely determined for every 1leq ileq n-1.

Background

The paper introduces fundamental equations for a Theta_n-contraction mathbf T=(T_1,ldots,T_n), generalizing the fundamental equations for Gamma_n-contractions. These equations involve tuples of operators (A_0^{(i)},ldots,A_p^{(i)}) acting on the defect space of T_n.

For Gamma_n-contractions, the corresponding fundamental operators are known to be uniquely determined. The paper explicitly asks whether the analogous fundamental-operator tuples for the broader class of Theta_n-contractions are uniquely determined and states that this problem remains unresolved.