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Three classes of smooth Banach submanifolds in B(E,F)

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Abstract

Let E,F be two Banach spaces, and B(E,F), Ф(E,F), SФ(E,F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result: Let Σ be any one of the following sets: {T ∈ Ф(E,F) : IndexT = const. and dim N(T) = const.}, {T ∈ SФ(E,F) : either dim N(T) = const. < ∞ or codim R(T) = const. < ∞} and {T ∈ R(E,F) : RankT =const.< ∞}. Then Σ is a smooth submanifold of B(E,F) with the tangent space T AΣ = {B ∈ B(E,F) : BN(A) ⊂ R(A)} for any A ∈ Σ. The result is available for the further application to Thom’s famous results on the transversility and the study of the infinite dimensional geometry.

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Authors and Affiliations

  1. Tseng Yaun Rong Functional Analysis Research Center, Harbin Normal University, Harbin, 150025, China

    Ji-pu Ma

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  1. Ji-pu Ma
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Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant No.10671049)

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Ma, Jp. Three classes of smooth Banach submanifolds in B(E,F). SCI CHINA SER A 50, 1233–1239 (2007). https://doi.org/10.1007/s11425-007-0104-6

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  • DOI: https://doi.org/10.1007/s11425-007-0104-6

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