A note on the dilation of a certain family of tetrablock contractions

We find an explicit tetrablock isometric dilation for every member $(A_{\alpha },B,P)$ of a family of tetrablock contractions indexed by a parameter $\alpha$ in the closed unit disc (only the first operator of the tetrablock contraction depends on the parameter). The dilation space is the same for any member of the family. Explicit dilation for the adjoint tetrablock contraction $(A_{\alpha }^{\ast },B^{\ast },P^{\ast })$ for every member of the family mentioned above is constructed as well. This example is important because it has been claimed in the literature that this example does not have a dilation. Taking cue from this construction and using Toeplitz operators on $H^2_{\mathbb D}(\mathcal D_P)$, we obtain necessary and sufficient conditions for a tetrablock contraction to have a certain type of tetrablock isometric dilation.