Advertisement

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
Article
  • 49 Downloads

Abstract

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 

Keywords

Closed range property coherent sheaves Fredholm theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Eschmeier J.: Are commuting systems of decomposable operators decomposable?. J. Oper. Theory 12, 213–219 (1984)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Eschmeier J.: On the essential spectrum of Banach space operators. Proc. Edinb. Math. Soc. 43, 511–528 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Eschmeier J.: Samuel multiplicity for several commuting operators. J. Oper. Theory 60, 399–414 (2008)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Eschmeier, J., Putinar, M.: Spectral Decompositions and Analytic Sheaves. London Mathematical Society Monographs, New Series, vol. 10, Clarendon Press, Oxford (1996)Google Scholar
  5. 5.
    Faas, D.: Zur Darstellungs- und Spektraltheorie für nichtvertauschende Operatortupel, Dissertation, Saarbrücken, 2008Google Scholar
  6. 6.
    Fang X.: The Fredholm index of a pair of commuting operators II. J. Funct. Anal. 256, 1669–1692 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grauert, H., Remmert, R.: Theorie der Steinschen Räume, Grundlehren der mathematischen Wissenschaften. vol. 227, Springer, Berlin (1977)Google Scholar
  8. 8.
    Kaballo W.: Holomorphe Semi-Fredholmfunktionen ohne komplementierte Kerne bzw. Bilder. Math. Nachrichten 91, 327–335 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. London Mathematical Society Monographs, vol. 20, The Clarendon Press, New York (2000)Google Scholar
  10. 10.
    Leiterer J.: Banach coherent analytic Fréchet sheaves. Math. Nachrichten 85, 91–109 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Mantlik, F.: Parameterabhängige lineare Gleichungen in Banach- und in Frécheträumen, Dissertation, Universität Dortmund, 1988Google Scholar
  12. 12.
    Markoe A.: Analytic families of differential complexes. J. Funct. Anal. 9, 181–188 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Miller T.L., Miller V.G., Neumann M.M.: The Kato-type spectrum and local spectral theory. Czech. Math. J. 57, 831–842 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Miller T.L., Müller V.: The closed range property for Banach space operators. Glasgow Math. J. 50, 17–26 (2008)zbMATHCrossRefGoogle Scholar
  15. 15.
    Northcott D.G.: Lessons on Rings, Modules and Multiplicities. Cambridge University Press, London (1968)zbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Fachrichtung MathematikUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations