Triangular Tetrablock-contractions, factorization of contractions, dilation and subvarieties

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\
Revised, 16 pages