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
Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B^+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B...Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B^+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B^+(E, F). In this paper we introduce an unbounded domain ?(A, A^+) in B(E, F) for A ∈ B^+(E, F) and A^+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A^+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A^+)^(-1) is the generalized inverse of T with R(B)=R(A^+) and N(B)=N(A^+), for T∈?(A, A^+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A^+,T) from ?(A,A^+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A^+) : R(T)∩ N(A^+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A^+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A^+). It seems to be useful for developing global analysis and geomatrical method in differential equations.展开更多
Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group an...Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group and also a smooth hypersurface in B（R^n） with the dimension n × n.展开更多
Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as a...Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as an algebra,while B(Rn,Rm) for m = n,is a Banach space but not an algebra;meanwhile,it is clear that R(E,F) is neither a Banach space nor an algebra.However,in this paper,it is proved that all of them have a common property in geometry and topology,i.e.,they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces).Let Σr be the set of all operators of finite rank r in B(E,F) (or B(Rn,Rm)).In fact,we have that 1) suppose Σr∈ B(Rn,Rm),and then Σr is a smooth and path-connected submanifold of B(Rn,Rm) and dimΣr = (n + m)r-r2,for each r ∈ [0,min{n,m});if m = n,the same conclusion for Σr and its dimension is valid for each r ∈ [0,min{n,m}];2) suppose Σr∈ B(E,F),and dimF = ∞,and then Σr is a smooth and path-connected submanifold of B(E,F) with the tangent space TAΣr = {B ∈ B(E,F) : BN(A)-R(A)} at each A ∈Σr for 0 r 【 ∞.The routine methods for seeking a path to connect two operators can hardly apply here.A new method and some fundamental theorems are introduced in this paper,which is development of elementary transformation of matrices in B(Rn),and more adapted and simple than the elementary transformation method.In addition to tensor analysis and application of Thom’s famous result for transversility,these will benefit the study of infinite geometry.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.
文摘Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B^+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B^+(E, F). In this paper we introduce an unbounded domain ?(A, A^+) in B(E, F) for A ∈ B^+(E, F) and A^+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A^+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A^+)^(-1) is the generalized inverse of T with R(B)=R(A^+) and N(B)=N(A^+), for T∈?(A, A^+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A^+,T) from ?(A,A^+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A^+) : R(T)∩ N(A^+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A^+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A^+). It seems to be useful for developing global analysis and geomatrical method in differential equations.
文摘Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group and also a smooth hypersurface in B（R^n） with the dimension n × n.
基金supported by National Natural Science Foundation of China (Grant Nos.10671049,10771101)
文摘Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as an algebra,while B(Rn,Rm) for m = n,is a Banach space but not an algebra;meanwhile,it is clear that R(E,F) is neither a Banach space nor an algebra.However,in this paper,it is proved that all of them have a common property in geometry and topology,i.e.,they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces).Let Σr be the set of all operators of finite rank r in B(E,F) (or B(Rn,Rm)).In fact,we have that 1) suppose Σr∈ B(Rn,Rm),and then Σr is a smooth and path-connected submanifold of B(Rn,Rm) and dimΣr = (n + m)r-r2,for each r ∈ [0,min{n,m});if m = n,the same conclusion for Σr and its dimension is valid for each r ∈ [0,min{n,m}];2) suppose Σr∈ B(E,F),and dimF = ∞,and then Σr is a smooth and path-connected submanifold of B(E,F) with the tangent space TAΣr = {B ∈ B(E,F) : BN(A)-R(A)} at each A ∈Σr for 0 r 【 ∞.The routine methods for seeking a path to connect two operators can hardly apply here.A new method and some fundamental theorems are introduced in this paper,which is development of elementary transformation of matrices in B(Rn),and more adapted and simple than the elementary transformation method.In addition to tensor analysis and application of Thom’s famous result for transversility,these will benefit the study of infinite geometry.
