Admissible fundamental operators(star)

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 Gamma-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 Gamma-contractions are then applied to tetrablock contractions to figure out when two pairs (F1, F2) and (G(1), G(2)) 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. (C) 2015 Elsevier Inc. All rights reserved.