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

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(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 ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.

Three classes of smooth Banach submanifolds in <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">E,F</Emphasis>)

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