##### Venue: Lecture Hall I, Department of Mathematics

We study homomorphisms $\rho_{V}$ given by $\rho_{V}(f)= \begin{pmatrix} f(w)I_n & \sum_{i=1}^{m} \partial_i f(w) V_{i} \\ 0 & f(w) I_n \end{pmatrix}$, $f \in \mathcal{O} (\Omega_{\mathbf{A}})$ defined on $\mathcal{O} (\Omega_{\mathbf{A}})$, where $\Omega_{\mathbf{A}}$ is a bounded domain of the form $\Omega_\mathbf A := \{ (z_1 ,z_2, \ldots, z_m) : | z_1 A_1 + \cdots + z_m A_m |_{\rm op} < 1 \}$ for some choice of a linearly independent set of $n\times n$ matrices $\{ A_1, \ldots, A_m \}.$

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