Let Gamma_3 be the closed symmetrized tridisc defined by
Gamma_3 = {(z_1 + z_2 + z_3; z_1z_2 + z_2z_3 + z_3z_1; z_1z_2z_3) : |z_i| <= 1; i = 1, 2, 3}
Let us denote by b Gamma_3, the distinguished boundary of Gamma_3 defined as
b Gamma_3 = {(z_1 + z_2 + z_3, z_1z_2 + z_2z_3 + z_3z_1, z_1z_2z_3) : |z_i|< = 1, i = 1, 2,3}
A 3-tuple of commuting operators (S_1,S_2,P) defined on a Hilbert space H for which
Gamma_3 is a spectral set is called a Gamma_3-contraction. In particular, when( S_1, S_2, P) are
commuting normal operators and have their Taylor joint spectrum sigma(S1; S2; P) contained in b Gamma_3
we call (S1, S2, P) a Gamma_3-unitary. In this talk, we show that a Gamma_3-contraction does not always have
a Gamma_3-unitary dilation. In other words, all Gamma_3-contractions do not have normal dilation to the distinguished
boundary b Gamma_3. To do that, we introduce the notion of fundamental operator equations for a 3-tuple of commuting
operators (S_1, S_2, P) on a Hilbert space H with || P ||<= 1, in the following way:
S_1- S_2^*P = D_PX_1D_P  and  S_2 - S _1^*P = D_PX_2D_P , where X_1,X_2 are bounded operators on Ran(I - P ^*P) closure
and D_P is the square root of (I-P^*P). We show that the fundamental equations have unique solutions when (S_1, S_2, P) is a
Gamma_3-contraction. For a Gamma_3-contraction (S_1, S_2, P), we pair up the unique solutions of the fundamental operator
equations as (F_1, F_2) and call it the fundamental operator pair of (S_1, S_2, P). We establish the fact that a Gamma_3-contraction
(S_1, S_2, P) has a normal dilation to the distinguished boundary b Gamma_3 only if F_1^* F1- F_1F_1^* = F_2^*F2 - F_2F_2^*. At the
end we produce an example of a finite dimensional Gamma_3-contraction whose fundamental operator pair fails to satisfy the above operator identity.