A fundamental result of B. Sz. Nazy states that every contraction T on a Hilbert space \({\mathcal {H}}\) has a unitary unitary dilation U to some Hilbert space \({\mathcal {K}}\) containing \({\mathcal {H}}\) as a subspace such that \(\begin{aligned} T^n=P_{{\mathcal {H}}}U^n_{|_{{\mathcal {H}}}}~{\textrm{for}}~n=1,2,\ldots , \end{aligned}\) where \(P_{{\mathcal {H}}}\) is a projection from \({\mathcal {K}}\) onto \({\mathcal {H}}.\) We can extend this theory sensibly to families of operators by considering the isometric dilation of a contraction on a Hilbert space. It is natural to ask whether this idea can be generalised, where the contraction T is substituted by a commuting n-tuple of operators \((S_1,\ldots , S_n)\) 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: \(\begin{aligned} E_1^*E_1-E_1E_1^*= E_2^*E_2-E_2E_2^*, \end{aligned}\) 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 described above. We develop an explicit \(\Gamma _3\) -isometric dilation for each member \((S_1^{(\alpha )}, S_2,S_3)\) of a family of \(\Gamma _3\) - contractions indexed by a parameter \(\alpha \) on the closed unit disc. We show that the dilation space is the same for any member of the family.
