Admissible fundamental operators associated with two domains related to $\mu$-synthesis

In this article, we discuss necessary condition of conditional dilation for both completely non-unitary (c.n.u) ${\mathrm{\Gamma }}_n$-contractions and c.n.u $\mathbb{E}$-contractions. Consider two tuples, $(A_1,\dots ,A_{n-1})$ and $(B_1,\dots ,B_{n-1})$, of operators defined on two Hilbert spaces. One of the principal goals is to identify a necessary and a sufficient condition guaranteeing the existence of a c.n.u ${\mathrm{\Gamma }}_n$-contraction $(S_1,\dots ,S_{n-1},P)$ such that $(A_1,\dots ,A_{n-1})$ becomes the ${\mathcal{F}}_O$-tuple of $(S_1,\dots ,S_{n-1},P)$ and $(B_1,\dots ,B_{n-1})$ becomes the ${\mathcal{F}}_O$-tuple of $(S_1^{\ast },\dots ,S_{n-1}^{\ast },P^{\ast })$. Also for given two pairs of operators $(F_1,F_2)$ and $(G_1,G_2)$ defined on two Hilbert spaces, we examine when there is a c.n.u $\mathbb{E}$-contraction $(A,B,P)$ such that $(F_1,F_2)$ becomes the ${\mathcal{F}}_O$-pair of $(A,B,P)$ and $(G_1,G_2)$ becomes the ${\mathcal{F}}_O$-pair of $(A^{\ast },B^{\ast },P^{\ast })$.