Admissible fundamental operators

Let F and G be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigates when there is a Γ-contraction (S,P) such that F is the fundamental operator of (S,P) and G is the fundamental operator of (S⁎,P⁎). Theorem 1 puts a necessary condition on F and G for them to be the fundamental operators of (S,P) and (S⁎,P⁎) respectively. Theorem 2 shows that this necessary condition is also sufficient provided we restrict our attention to a certain special case. The general case is investigated in Theorem 3. Some of the results obtained for Γ-contractions are then applied to tetrablock contractions to figure out when two pairs (F1,F2) and (G1,G2) acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction (A,B,P) and its adjoint (A⁎,B⁎,P⁎) respectively. This is the content of Theorem 3.