Representation of Operators on Spaces of Holomorphic Functions in ${\mathbb{C}}^n$

We investigate operators between spaces of holomorphic functions in several complex variables. Let $G_1,G_2\subset {\mathbb{C}}^n$ be cylindrical domains. We construct a canonical map from the space of bounded linear operators $\mathcal{L}(H(G_1),H(G_2))$ to $H(G_1^b\times G_2)$ and prove that it is a topological isomorphism (Theorem~\ref{pierwsze twierdzenie}). We then establish uniform estimates for operators on bounded, complete $n$-circled domains (Theorem~\ref{thm:4.8}) and show that sequences of operators on smaller domains satisfying suitable uniform bounds uniquely determine a global operator (Theorem~\ref{thm:4.9}). Together, these results provide a unified framework for representing and extending operators on spaces of holomorphic functions in several complex variables.