Schwarz lemma at the boundary on the matrix ball of type $\mathcal{I}$ in $\mathbb{C}^n[m\times m]$

In this talk, following a brief historical overview, we present a boundary version of the Schwarz lemma in $\mathbb{C}^n$, together with its extension to the matrix ball of the first type in the space $\mathbb{C}^n[m \times m]$.