  A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed
tetrablock $\bar{\mathbb E}$ is a spectral set, is called a
\textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction},
where \[ \mathbb E=\{(a_{11},a_{22}, \det A):\, A=[a_{ij}]\in \mathcal
M_2(\mathbb C), \; \|A\| <1 \} \subset \mathbb C^3 \] is a polynomially convex
domain which is naturally associated with the $\mu$-synthesis problem. We
introduce triangular $\mathbb E$-contractions and prove that every pure
triangular $\mathbb E$-contraction dilates to a pure triangular $\mathbb
E$-isometry. We construct a functional model for a pure triangular $\mathbb
E$-isometry and apply that model to find a new proof for the famous
Berger-Coburn-Lebow Model Theorem for commuting isometries. Next we give an
alternative proof to the more generalized version of Berger-Coburn-Lebow Model,
namely the factorization of a pure contraction due to Das, Sarkar and Sarkar
(\textit{Adv. Math.} 322 (2017), 186 -- 200). We find a necessary and
sufficient condition for the existence of $\mathbb E$-unitary dilation of an
$\mathbb E$-contraction $(A,B,P)$ on the smallest dilation space and show that
it is equivalent to the existence of a distinguished variety in $\mathbb E$
when the defect space $D_{P^*}$ is finite dimensional.

Functional Analysis;Complex Variables