There have been many really positive results co ncerning the weakly compact operators on Banach lattices in terms of their order structure as well as in many respects. This paper will survey some known recent result...There have been many really positive results co ncerning the weakly compact operators on Banach lattices in terms of their order structure as well as in many respects. This paper will survey some known recent results in this area.展开更多
Some characterizations of preregular operators between two Banach lattices are presented. Then several sufficient conditions for preregular operators being regular are given, and some related results are also obtained.
This paper deals with a characterization for a Banach lattice in which the lattice operations are weakly sequentially continuous. As an application an elementary proof for an important result due to Wickstead is pro...This paper deals with a characterization for a Banach lattice in which the lattice operations are weakly sequentially continuous. As an application an elementary proof for an important result due to Wickstead is provided.展开更多
In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality：∑i=1,aib（m＋1-i）≤∑i=1maibp（i）≤∑i=1,aibi,in which the real sequences {ai}i and {bi}i are in increasing order, and p（i） indicates a ...In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality：∑i=1,aib（m＋1-i）≤∑i=1maibp（i）≤∑i=1,aibi,in which the real sequences {ai}i and {bi}i are in increasing order, and p（i） indicates a random permutation. We now consider a sequence in lp with 1 〈 p 〈 ∞, and a sequence in a Banach lattice X. Instead of normal multiplication, we consider the tensor product of lp and X. We show that in Wittstock injective tensor product, lp iX, and Fremlin projective tensor product, lp FX, the rearrangement inequality still exists.展开更多
By means of the absolute continuities of the abstract functions whose values lie in X, the characterizations of Banach lattices X being 1-concave or satisfying lower p-estimates are obtained.
Hudzik and Kurc discussed some best approximation problems in Banach lattices by means of monotonicities. This paper deals with more general best approximation problems in Banach lattices. Existence, uniqueness, stabi...Hudzik and Kurc discussed some best approximation problems in Banach lattices by means of monotonicities. This paper deals with more general best approximation problems in Banach lattices. Existence, uniqueness, stability and continuity for such best approximation problems are discussed.展开更多
We present here that F（E,F）, the space of all r-compact operators from E into F, is a generalised sublattice of L^r（E, F） for arbitary Banach lattices E and F, and that the characterization of the regular norm on ...We present here that F（E,F）, the space of all r-compact operators from E into F, is a generalised sublattice of L^r（E, F） for arbitary Banach lattices E and F, and that the characterization of the regular norm on F（E, F） is order continuous. Some conditions for F（E, F） to be a KB-space or a band in .L（E, F） are also provided.展开更多
This article is committed to deal with measure of non-compactness of operators in Banach spaces.Firstly,the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the ua...This article is committed to deal with measure of non-compactness of operators in Banach spaces.Firstly,the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication)is a normed semigroup,and the mapping J from C(X)onto F(Ω)is a fully order-preserving positively linear surjective isometry,whereΩis the closed unit ball of X^*and F(Ω)the collection of all continuous and w^*-lower semicontinuous sublinear functions on X^*but restricted toΩ.Furthermore,both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC.The quotient space EC/EK is an abstract M space,hence,order isometric to a sublattice of C(K)for some compact Haudorspace K,and(FQJ)C which is a closed cone is contained in the positive cone of C(K),where Q:EC→EC/EK is the quotient mapping and F:EC/EK→C(K)is a corresponding order isometry.Finally,the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X,thenμ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K),∀T∈B(X).展开更多
文摘There have been many really positive results co ncerning the weakly compact operators on Banach lattices in terms of their order structure as well as in many respects. This paper will survey some known recent results in this area.
文摘Some characterizations of preregular operators between two Banach lattices are presented. Then several sufficient conditions for preregular operators being regular are given, and some related results are also obtained.
文摘This paper deals with a characterization for a Banach lattice in which the lattice operations are weakly sequentially continuous. As an application an elementary proof for an important result due to Wickstead is provided.
文摘In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality：∑i=1,aib（m＋1-i）≤∑i=1maibp（i）≤∑i=1,aibi,in which the real sequences {ai}i and {bi}i are in increasing order, and p（i） indicates a random permutation. We now consider a sequence in lp with 1 〈 p 〈 ∞, and a sequence in a Banach lattice X. Instead of normal multiplication, we consider the tensor product of lp and X. We show that in Wittstock injective tensor product, lp iX, and Fremlin projective tensor product, lp FX, the rearrangement inequality still exists.
文摘By means of the absolute continuities of the abstract functions whose values lie in X, the characterizations of Banach lattices X being 1-concave or satisfying lower p-estimates are obtained.
基金Supported by National Natural Science Foundation of China, Grant (10471032)
文摘Hudzik and Kurc discussed some best approximation problems in Banach lattices by means of monotonicities. This paper deals with more general best approximation problems in Banach lattices. Existence, uniqueness, stability and continuity for such best approximation problems are discussed.
文摘We present here that F（E,F）, the space of all r-compact operators from E into F, is a generalised sublattice of L^r（E, F） for arbitary Banach lattices E and F, and that the characterization of the regular norm on F（E, F） is order continuous. Some conditions for F（E, F） to be a KB-space or a band in .L（E, F） are also provided.
基金The project supported in part by the National Natural Science Foundation of China(11801255)。
文摘This article is committed to deal with measure of non-compactness of operators in Banach spaces.Firstly,the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication)is a normed semigroup,and the mapping J from C(X)onto F(Ω)is a fully order-preserving positively linear surjective isometry,whereΩis the closed unit ball of X^*and F(Ω)the collection of all continuous and w^*-lower semicontinuous sublinear functions on X^*but restricted toΩ.Furthermore,both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC.The quotient space EC/EK is an abstract M space,hence,order isometric to a sublattice of C(K)for some compact Haudorspace K,and(FQJ)C which is a closed cone is contained in the positive cone of C(K),where Q:EC→EC/EK is the quotient mapping and F:EC/EK→C(K)is a corresponding order isometry.Finally,the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X,thenμ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K),∀T∈B(X).
