부산대학교 도서관

Author's summary: ``Let $G$ be a complex domain, $X$ and $Y$ Banach spaces and $A\colon G\rightarrow L(X,Y)$ holomorphic with $\text{Ker}\,A(\lambda)$, $\text{Im}\,A(\lambda)$ complemented, $\lambda\in G$. It is shown that the following conditions are equivalent: (1) $A$ has a holomorphic relative inverse on $G$; (2) the function $\lambda\mapsto\text{Ker}\,A(\lambda)$ is locally holomorphic on $G$; (3) the function $\lambda\mapsto\text{Im}\,A(\lambda)$ is locally holomorphic on $G$. Based on this, it is shown that a semi-Fredholm-valued holomorphic function $A$ has a holomorphic relative inverse on $G$ if and only if $\dim\text{Ker}\,A(\lambda)$ [codim $\text{Im}\,A(\lambda)$, respectively] is constant on $G$. The latter result is a generalization of the well-known result of G. R. Allan [J. London Math. Soc. {\bf 42} (1967), 509--513; MR0215097 (35 \#5940)] on one-sided holomorphic inverses.''