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 ...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:LetΣbe 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.<∞}.ThenΣis 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.展开更多
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...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 = ? n , 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(? n ) with dimΣ r = 2nr × r 2. Then a new geometrical construction of B(? n ), analogous to B +(E, F), is given besides its analysis and algebra constructions known well.展开更多
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.10671049)
文摘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:LetΣbe 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.<∞}.ThenΣis 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.
基金supported by National Natural Science Foundation of China (Grant No.10771101,10671049)
文摘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 = ? n , 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(? n ) with dimΣ r = 2nr × r 2. Then a new geometrical construction of B(? n ), analogous to B +(E, F), is given besides its analysis and algebra constructions known well.
