Integral Equations and Operator Theory

, Volume 67, Issue 3, pp 365–375 | Cite as

Closed Range Property for Holomorphic Semi-Fredholm Functions

  • Jörg Eschmeier
  • Dominik Faas


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.

Mathematics Subject Classification (2000)

Primary 47A13 32C35 Secondary 47A11 47A53 


Closed range property coherent sheaves Fredholm theory 


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© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Fachrichtung MathematikUniversität des SaarlandesSaarbrückenGermany

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