Closed Range Property for Holomorphic Semi-Fredholm Functions

Given Banach spaces X and Y, we show that, for each operator-valued analytic map ${alpha in mathcal O (D,mathcal L(Y,X))}$ satisfying the finiteness condition ${dim (X/alpha (z)Y) < infty}$ pointwise on an open set D in ${mathbb {C}^n}$ , the induced multiplication operator ${mathcal O(U,Y) stackrel{alpha}{longrightarrow} mathcal O (U,X)}$ has closed range on each Stein open set ${U subset D}$ . As an application we deduce that the generalized range ${{rm R}^{infty}(T) = bigcap_{k geq 1}sum_{| alpha | = k} T^{alpha}X}$ of a commuting multioperator ${T in mathcal L(X)^n}$ with ${dim(X/sum_{i=1}^n T_iX) < infty}$ can be represented as a suitable spectral subspace.