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

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.