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

Abstract: 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.