Admissible fundamental operators

Abstract

Let $F$ and $G$ be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigate 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 $(S^{\ast },P^{\ast })$. Theorem 1 puts a necessary condition on $F$ and $G$ for them to be the fundamental operators of $(S,P)$ and $(S^{\ast },P^{\ast })$ respectively. Theorem 2 shows that this necessary condition is sufficient too 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 $(F_1,F_2)$ and $(G_1,G_2)$ acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction $(A,B,P)$ and its adjoint $(A^{\ast },B^{\ast },P^{\ast })$ respectively. This is the content of Theorem 4.

References in corpus (3)

• Operator theory on the tetrablock
• From Stinespring dilation to Sz.-Nagy dilation on the symmetrized bidisc and operator models
• A Note on Tetrablock Contractions