Admissible fundamental operators

TL;DR

This paper characterizes when two bounded operators can serve as fundamental operators for a $\Gamma$-contraction and its adjoint, providing necessary and sufficient conditions and extending results to tetrablock contractions.

Contribution

It establishes necessary and sufficient conditions for fundamental operators of $\Gamma$-contractions and applies these results to tetrablock contractions, advancing the understanding of their operator structure.

Findings

Necessary condition for fundamental operators of $\Gamma$-contractions.

Sufficient condition in a special case for these operators.

Extension of results to tetrablock contractions.

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 $(S^*,P^*)$. 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 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 $(A^*, B^*, P^*)$ respectively. This is the content of Theorem 4.