If $T$ is a cnu (completely non-unitary) contraction on a Hilbert space, then its Nagy-Foias characteristic function is an operator valued analytic function on the unit disc $\mathbb{D}$ which is a complete invariant for the unitary equivalence class of $T$. $T$ is said to be homogeneous if $\varphi(T)$ is unitarily equivalent to $T$ for all elements $\varphi$ of the group $M$ of biholomorphic maps on $\mathbb{D}$. A stronger notion is of an associator. $T$ is an associator if there is a projective unitary representation $\sigma$ of $M$ such that $\varphi(T) = \sigma(\varphi)^* T \sigma(\varphi)$ for all $\varphi$ in $M$. In this talk we shall discuss the following result from a recent work of the speaker with G. Misra and S. Hazra: A cnu contraction is an associator if and only if its characteristic function $\theta$ has the factorization $\theta(z) = \pi_* (\varphi_z) C \pi(\varphi_z), z \in \mathbb{D}$ for two projective unitary representations $\pi, \pi_∗$ of $M$.

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Last updated: 13 Jun 2024