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

In this article, we discuss necessary condition of conditional dilation for both completely non-unitary (c.n.u) $Γ_{n}$-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 $Γ_n$-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 $(S_1^*,\dots, S_{n-1}^*,P^*)$. 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 $(A^*,B^*,P^*)$.