Homomorphisms of infinitely generated analytic sheaves

We prove that every homomorphism ${\mathcal{O}}_{\zeta }^E\to {\mathcal{O}}_{\zeta }^F$, with $E$ and $F$ Banach spaces and $\zeta \in {\mathbb{C}}^m$, is induced by a $(E,F)$-valued holomorphic germ, provided that $1\leq m<\infty$. A similar structure theorem is obtained for the homomorphisms of type ${\mathcal{O}}_{\zeta }^E\to {\mathcal{S}}_{\zeta }$, where ${\mathcal{S}}_{\zeta }$ is a stalk of a coherent sheaf of positive ${\mathfrak{m}}_{\zeta }$-depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert-Patyi.