The projective tensor product 2⊗ X of 2 with any Banach space X sits inside the space Rad(X) of a... more The projective tensor product 2⊗ X of 2 with any Banach space X sits inside the space Rad(X) of all almost unconditionally summable sequences in X. If X is of cotype 2 and u : X −→ Y is 2-summing, then u takes Rad(X) into 2⊗ Y. Consequently, if X is of cotype 2, then every operator from X to 2 is 1-summing if and only if 1⊗ X ⊆ 2⊗ X. In this case, each 2-summing operator from 2 to X is nuclear, and X does not have non-trivial type provided that dim X = ∞.
Commentationes Mathematicae Universitatis Carolinae
In this paper, we show the representation of Köthe dual of Banach sequence spaces ℓp [X] (1 ≤ p <... more In this paper, we show the representation of Köthe dual of Banach sequence spaces ℓp [X] (1 ≤ p < ∞) and give a characterization of that the spaces ℓp[X] (1 < p < ∞) are Grothendieck spaces.
Let $X$ and $Y$ be Banach spaces such that $X$ has a boundedly complete basis. Then $X\hat{\otime... more Let $X$ and $Y$ be Banach spaces such that $X$ has a boundedly complete basis. Then $X\hat{\otimes}Y$, the projective tensor product of $X$ and $Y$, has the Radon-Nikodym property (resp.~the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of $c_0$) if and only if $Y$ has the same property.
Let H : M m → M m be a holomorphic function of the algebra M m of complex m × m matrices. Suppose... more Let H : M m → M m be a holomorphic function of the algebra M m of complex m × m matrices. Suppose that H is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of H consists of zero trace elements, or there is a scalar sequence {λ n } and an invertible S in M m such that
ABSTRACT Let E be a Banach space with 1-unconditional basis. Denote by 𝛥(⊗ ^ n,π E) (resp. 𝛥(⊗ ^ ... more ABSTRACT Let E be a Banach space with 1-unconditional basis. Denote by 𝛥(⊗ ^ n,π E) (resp. 𝛥(⊗ ^ n,s,π E)) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by 𝛥(⊗ ^ n,|π| E) (resp. 𝛥(⊗ ^ n,s,|π| E)) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to E [n] , the completion of the n-concavification of E. Using these isometries, we also show that the norm of any (vector valued) continuous orthogonally additive homogeneous polynomial on E equals the norm of its associated symmetric linear operator.
Let X be a Banach space and 1 < p, p < ∞ such that 1/p + 1/p = 1. Then L p [0, 1]⊗X, respectively... more Let X be a Banach space and 1 < p, p < ∞ such that 1/p + 1/p = 1. Then L p [0, 1]⊗X, respectively L p [0, 1]⊗X, the projective, respectively injective, tensor product of L p [0, 1] and X, is a Grothendieck space if and only if X is a Grothendieck space and each continuous linear operator from L p [0, 1], respectively L p [0, 1], to X * , respectively X * * , is compact.
ABSTRACT Let φ be an Orlicz function that has a complementary function φ* and let ℓφ be an Orlicz... more ABSTRACT Let φ be an Orlicz function that has a complementary function φ* and let ℓφ be an Orlicz sequence space. We prove two results in this paper. Result 1: $$\ell_\varphi\hat{\otimes}_F X$$ , the Fremlin projective tensor product of ℓφ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓφ and X have the Radon-Nikodym property. Result 2: $$\ell_\varphi\tilde{\otimes}_i X$$ , the Wittstock injective tensor product of ℓφ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓφ and X have the Radon-Nikodym property and each positive continuous linear operator from hφ* to X is compact.
We investigate the relationship between the diagonal of the Fremlin projective tensor product of ... more We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice E with itself and the 2-concavification of E. Date: September 27, 2011. 2010 Mathematics Subject Classification. Primary: 46B42. Secondary: 46M05, 46B40, 46B45. Key words and phrases. Banach lattice, Fremlin projective tensor product, diagonal of tensor square, square of a Banach lattice, concavification.
ABSTRACT For Banach lattices and a Banach space Y, we compare the class of multiple p-summing n-l... more ABSTRACT For Banach lattices and a Banach space Y, we compare the class of multiple p-summing n-linear operators from to Y, to the class of positive multiple p-summing n-linear operators and the class of multiple p-concave n-linear operators for , and derive some inclusion properties. In particular, if F is a dual AL-space, then coincides with the space of all regular n-linear operators from to F.
The projective tensor product 2⊗ X of 2 with any Banach space X sits inside the space Rad(X) of a... more The projective tensor product 2⊗ X of 2 with any Banach space X sits inside the space Rad(X) of all almost unconditionally summable sequences in X. If X is of cotype 2 and u : X −→ Y is 2-summing, then u takes Rad(X) into 2⊗ Y. Consequently, if X is of cotype 2, then every operator from X to 2 is 1-summing if and only if 1⊗ X ⊆ 2⊗ X. In this case, each 2-summing operator from 2 to X is nuclear, and X does not have non-trivial type provided that dim X = ∞.
Commentationes Mathematicae Universitatis Carolinae
In this paper, we show the representation of Köthe dual of Banach sequence spaces ℓp [X] (1 ≤ p <... more In this paper, we show the representation of Köthe dual of Banach sequence spaces ℓp [X] (1 ≤ p < ∞) and give a characterization of that the spaces ℓp[X] (1 < p < ∞) are Grothendieck spaces.
Let $X$ and $Y$ be Banach spaces such that $X$ has a boundedly complete basis. Then $X\hat{\otime... more Let $X$ and $Y$ be Banach spaces such that $X$ has a boundedly complete basis. Then $X\hat{\otimes}Y$, the projective tensor product of $X$ and $Y$, has the Radon-Nikodym property (resp.~the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of $c_0$) if and only if $Y$ has the same property.
Let H : M m → M m be a holomorphic function of the algebra M m of complex m × m matrices. Suppose... more Let H : M m → M m be a holomorphic function of the algebra M m of complex m × m matrices. Suppose that H is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of H consists of zero trace elements, or there is a scalar sequence {λ n } and an invertible S in M m such that
ABSTRACT Let E be a Banach space with 1-unconditional basis. Denote by 𝛥(⊗ ^ n,π E) (resp. 𝛥(⊗ ^ ... more ABSTRACT Let E be a Banach space with 1-unconditional basis. Denote by 𝛥(⊗ ^ n,π E) (resp. 𝛥(⊗ ^ n,s,π E)) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by 𝛥(⊗ ^ n,|π| E) (resp. 𝛥(⊗ ^ n,s,|π| E)) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to E [n] , the completion of the n-concavification of E. Using these isometries, we also show that the norm of any (vector valued) continuous orthogonally additive homogeneous polynomial on E equals the norm of its associated symmetric linear operator.
Let X be a Banach space and 1 < p, p < ∞ such that 1/p + 1/p = 1. Then L p [0, 1]⊗X, respectively... more Let X be a Banach space and 1 < p, p < ∞ such that 1/p + 1/p = 1. Then L p [0, 1]⊗X, respectively L p [0, 1]⊗X, the projective, respectively injective, tensor product of L p [0, 1] and X, is a Grothendieck space if and only if X is a Grothendieck space and each continuous linear operator from L p [0, 1], respectively L p [0, 1], to X * , respectively X * * , is compact.
ABSTRACT Let φ be an Orlicz function that has a complementary function φ* and let ℓφ be an Orlicz... more ABSTRACT Let φ be an Orlicz function that has a complementary function φ* and let ℓφ be an Orlicz sequence space. We prove two results in this paper. Result 1: $$\ell_\varphi\hat{\otimes}_F X$$ , the Fremlin projective tensor product of ℓφ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓφ and X have the Radon-Nikodym property. Result 2: $$\ell_\varphi\tilde{\otimes}_i X$$ , the Wittstock injective tensor product of ℓφ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓφ and X have the Radon-Nikodym property and each positive continuous linear operator from hφ* to X is compact.
We investigate the relationship between the diagonal of the Fremlin projective tensor product of ... more We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice E with itself and the 2-concavification of E. Date: September 27, 2011. 2010 Mathematics Subject Classification. Primary: 46B42. Secondary: 46M05, 46B40, 46B45. Key words and phrases. Banach lattice, Fremlin projective tensor product, diagonal of tensor square, square of a Banach lattice, concavification.
ABSTRACT For Banach lattices and a Banach space Y, we compare the class of multiple p-summing n-l... more ABSTRACT For Banach lattices and a Banach space Y, we compare the class of multiple p-summing n-linear operators from to Y, to the class of positive multiple p-summing n-linear operators and the class of multiple p-concave n-linear operators for , and derive some inclusion properties. In particular, if F is a dual AL-space, then coincides with the space of all regular n-linear operators from to F.
Uploads
Papers by Qingying Bu