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On holomorphic families of subspaces of a banach space

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Abstract

In this paper we study holomorphic families of subspaces of a Banach space, which are parametrized by some analytic space. We consider the question of the existence of a holomorphic complement for a given family. It turns out that such a complement does always exist if the basic space is a Stein space, provided for each fixed value of the parameter the subspaces of the family are complemented. From this follows, in particular, the existence of a holomorphic left inverse of a holomorphic operator function, provided such a left inverse exists for each fixed value of the parameter. The latter result gives a positive answer to a question formulated by I.C. Gohberg in a personal conversation with the author.

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References

  1. R.C. Gunning, H. Rossi: Analytic functions of several complex variables. Prentice Hall 1965.

  2. H. Grauert: Analytische Faserungen über holomorph vollständigen Räumen. Math. Ann. 135 (1958), 263–273.

    Google Scholar 

  3. A.L. Onishchik: Complex analytic bundle spaces. in: 4th mathematical summer school, Katsiveli 1966, Kiev, "Naukova dumka" (1968), 192–233 [in Russian].

  4. L. Bungart: On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers. Topology 7 (1968), 55–68.

    Google Scholar 

  5. I.C. Gohberg, A.S. Markus: Two theorems on the gap between subspaces of a Banach space. Uspekhi matem.nauk. 14:5 (1959), 135–140 [in Russian].

    Google Scholar 

  6. A.S. Markus: On some properties of linear operators, connected with the notion of the gap. Kishinev, Uchenye ZAPISKI Universiteta 39 (1959), 265–272 [in Russian].

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Authors
  1. M. A. Shubin
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Additional information

Note of the Editor: This paper was originally published in "Mat. Issled. (Kishinev) 5, vyp 4 (18) (1970), 153–165”. The editor is grateful to K. Clancey, W. Kaballo and G. Ph. A. Thijsse for preparing the translation from Russian.

Note of the Editor: After the appearance of the original version of the above paper the author informed the editor of Mat. Issled (see Mat. Issled. (Kishinev) 6, vyp (19) (1971), 180) that Corollaries 2 and 3 of Theorem 2 are contained in the papers of G.R. Allan, J. London Math. Soc. 42 (1967), 463–470 and 509–513.

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Shubin, M.A. On holomorphic families of subspaces of a banach space. Integr equ oper theory 2, 407–420 (1979). https://doi.org/10.1007/BF01682677

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  • DOI: https://doi.org/10.1007/BF01682677

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