The smooth Banach submanifold B*(E, F) in B(E, F)

Abstract

Let E, F be two Banach spaces, B(E, F),B ⁺(E, F), Φ(E, F), SΦ(E, F) and R(E, F) be bounded linear, double splitting, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. Let Σ be any one of the following sets: {T ∈ Φ(E, F): Index T = constant and dim N(T) = constant}, {T ∈ SΦ(E, F): either dim N(T) =constant< ∞ or codim R(T) =constant< ∞} and {T ∈ R(E, F): Rank T =constant< ∞}. Then it is known that gS is a smooth submanifold of B(E, F) with the tangent space T _A Σ = {B ∈ B(E, F): BN(A) ⊂ R(A)} for any A ∈ Σ. However, for B*(E, F) = {T ∈ B ⁺(E, F): dimN(T) = codimR(T) = ∞} without the characteristic numbers, dimN(A), codimR(A), index(A) and Rank(A) of the equivalent classes above, it is very difficult to find which class of operators in B*(E, E) forms a smooth submanifold of B(E, F). Fortunately, we find that B*(E, F) is just a smooth submanifold of B(E, F) with the tangent space T _A B*(E, F) = {T ∈ B(E, F): TN(A) ⊂ R(A)} for each A ∈ B*(E, F). Thus the geometric construction of B ⁺(E, F) is obtained, i.e., B ⁺(E, F) is a smooth Banach submanifold of B(E, F), which is the union of the previous smooth submanifolds disjoint from each other. Meanwhile we give a smooth submanifold S(A) of B(E, F), modeled on a fixed Banach space and containing A for any A ∈ B ⁺(E, F). To end these, results on the generalized inverse perturbation analysis are generalized. Specially, in the case E = F = ℝⁿ, it is obtained that the set Σ _r of all n × n matrices A with Rank(A) = r < n is an arcwise connected and smooth hypersurface (submanifold) in B(ℝⁿ) with dimΣ _r = 2nr × r ². Then a new geometrical construction of B(ℝⁿ), analogous to B ⁺(E, F), is given besides its analysis and algebra constructions known well.