Necessary conditions for existence of $\Gamma_n$-contractions and examples of $\Gamma_3$-contractions

TL;DR

This paper investigates the conditions under which a tuple of operators called $\Gamma_n$-contractions can be dilated to a $\Gamma_n$-isometry, providing necessary conditions and examples that challenge previous assumptions about sufficiency.

Contribution

It establishes necessary conditions for $\Gamma_n$-isometric dilations and presents examples showing these conditions are not sufficient in general.

Findings

Derived necessary conditions for $\Gamma_n$-isometric dilation.

Provided an example of a $\Gamma_3$-contraction with a dilation that does not satisfy certain conditions.

Showed that sufficient conditions for dilation are not always necessary, even with special operator structures.

Abstract

The fundamental result of B. Sz. Nazy states that every contraction has a coisometric extension and a unitary dilation. The isometric dilation of a contraction on a Hilbert space motivated whether this theory can be extended sensibly to families of operators. It is natural to ask whether this idea can be generalized, where the contraction $T$ is substituted by a commuting $n$-tuples of operators $(S_1,\cdots, S_n)$ acting on some Hilbert space having $\Gamma_n$ as a spectral set. We derive the necessary conditions for the existence of a $\Gamma_n$-isometric dilation for $\Gamma_n$-contractions. Also we discuss an example of a $\Gamma_3$-contraction $(S_1, S_2, S_3)$ acting on some Hilbert space $\mathcal H,$ which has a $\Gamma_3$-isometric dilation, but it fails to satisfy the following condition: $$E_1^*E_1-E_1E_1^*= E_2^*E_2-E_2E_2^*,$$ where $E_1$ and $E_2$ are the fundamental operators of $(S_1, S_2, S_3),$ $(S_1,S_2)$ is a pair of commuting contractions and $S_3$ is a partial isometry. Thus, the set of sufficient conditions for the existence of a $\Gamma_3$-isometric dilation breaks down, in general, to be necessary, even when the $\Gamma_3$-contraction $(S_1, S_2, S_3)$ has the special structure as described above.