The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a modu...The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(μ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.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
In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.
By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it...By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.展开更多
First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact...First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T^+,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.展开更多
In this paper,by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk,we give a proof of the Beurling type theorem on the Bergman space of Aleman,Richter and Sundb...In this paper,by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk,we give a proof of the Beurling type theorem on the Bergman space of Aleman,Richter and Sundberg(1996) via the Hardy space of the bidisk.展开更多
Intuitively, non-smooth points might look like exposed points. However, in this paper we show that real Banach spaces having dimension greater than or equal to three can be equivalently renormed to obtain non-smooth p...Intuitively, non-smooth points might look like exposed points. However, in this paper we show that real Banach spaces having dimension greater than or equal to three can be equivalently renormed to obtain non-smooth points which are also non-exposed.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 10871016)
文摘The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(μ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.
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.
基金supported by Research Foundation for Doctor Programme (Grant No. 20060055010)National Natural Science Foundation of China (Grant No. 10871101)
文摘In this paper we introduce the isometric extension problem of isometric mappings between two unit spheres. Some important results of the related problems are outlined and the recent progress is mentioned.
基金Supported by the National Natural Science Foundation of China (Grant No. 10471114)
文摘By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.
基金the State Committee for Scientific Research,Poland (Grant No.1P03A1127)the National Nature Science Foundation of China (Grant Nos.10471032,10671049)
文摘First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T^+,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.
基金supported by National Natural Science Foundation of China (Grant No. 10871083)National Science Foundation of USA (Grant No. 0457285)
文摘In this paper,by lifting the Bergman shift as the compression of an isometry on a subspace of the Hardy space of the bidisk,we give a proof of the Beurling type theorem on the Bergman space of Aleman,Richter and Sundberg(1996) via the Hardy space of the bidisk.
文摘Intuitively, non-smooth points might look like exposed points. However, in this paper we show that real Banach spaces having dimension greater than or equal to three can be equivalently renormed to obtain non-smooth points which are also non-exposed.
