A product formula for homogeneous characteristic functions

A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $varphi(T)$ is unitarily equivalent to $T$ for all $varphi$ in the group M{o}b of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $sigma$ of M{o}b is said to be associated with an operator T if $varphi(T)= sigma(varphi)^star T sigma(varphi)$ for all $varphi$ in M{o}b. In this paper, we develop a M{o}bius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $sigma$, then there is a unique projective unitary representation $hat{sigma}$, extending $sigma$, associated with the minimal unitary dilation of $T$. The representation $hat{sigma}$ is given in terms of $sigma$ by the formula $$ hat{sigma} = (pi otimes D_1^+) oplus sigma oplus (pi_star otimes D_1^-), $$ where $D_1^pm$ are the two Discrete series representations (one holomorphic and the other anti-holomorphic) living on the Hardy space $H^2(mathbb D)$, and $pi, pi_star$ are representations of M{o}b living on the two defect spaces of $T$ defined explicitly in terms of $sigma$. Moreover, a cnu contraction $T$ has an associated representation if and only if its Sz.-Nagy--Foias characteristic function $theta_T$ has the product form $theta_T(z) = pi_star(varphi_z)^* theta_T(0) pi(varphi_z),$ $zin mathbb D$, where $varphi_z$ is the involution in M{o}b mapping $z$ to $0.$ We obtain a concrete realization of this product formula %the two representations $pi_star$ and $pi$ for a large subclass of homogeneous cnu contractions from the Cowen-Douglas class.