Bases In Spaces Of Regular Multilinear Operators And Homogeneous Polynomials On Banach Lattices
Bases In Spaces Of Regular Multilinear Operators And Homogeneous Polynomials On Banach Lattices
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发表时间：
2018
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通讯作者：
K. Navoyan
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作者：
K. Navoyan
对于Banach格E1，. . .，Em和F的1-无条件基，证明了单项式序列构成L（E1，. . .，Em;F），从E1×···×Em到F的所有正则m-线性算子的Banach格，当且仅当E1，. . .，Em是收缩的，从E1×· ··×Em到F的每一个正m-线性算子都是弱序列连续的.作为结果，我们得到了m重Fremlin射影张量积E1 ≠ 0的充要条件|π| · · · ⊗̂| π| Em（resp. m重正内射张量积E_1 ≠ E_2|| · · · ⊗̌||Em）有收缩基或有界完备基。对于具有1-无条件基的Banach格E和F，证明了单项式序列构成从E到F的所有正则非齐次多项式的Banach格P（E;F）的1-无条件基的充要条件是E有一个收缩基，并且从E到F的每个正的m-齐次多项式都是弱序列连续的.作为结果，我们得到了m重对称正投射张量积m，s，|π| E（分别为m重对称正内射张量积m，s，||E）有收缩基或有界完备基。对于向量格E，n ∈ N，设E是E的n重Fremlin向量格对称张量积.对m，n ∈ N，且m > n，我们证明了：（i）若k ∈ m，sE是一致完备的，则k ∈ n，sE正同构于k ∈ m，sE的一个补子空间;（ii）若存在φ ∈ E k+，使得ker（φ）是E中的一个投影带，则k ∈ n，sE格同构于k ∈ m，sE的一个投影带.对于n重Fremlin Banach格对称张量积ε n，s，我们也得到了类似的结果，|π|其中E是Banach格。
For Banach lattices E1, . . . , Em and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of L(E1, . . . , Em;F ), the Banach lattice of all regular m-linear operators from E1×· · ·×Em to F , if and only if each basis of E1, . . . , Em is shrinking and every positive m-linear operator from E1×· · ·×Em to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the mfold Fremlin projective tensor product E1⊗̂|π| · · · ⊗̂|π|Em (resp. the m-fold positive injective tensor product E1⊗̌| | · · · ⊗̌| |Em) has a shrinking basis or a boundedly complete basis. For Banach lattices E and F with 1-unconditional bases, we show that the monomial sequence forms a 1-unconditional basis of P(E;F ), the Banach lattice of all regular mhomogeneous polynomials from E to F , if and only if E has a shrinking basis and every positive m-homogeneous polynomial from E to F is weakly sequentially continuous. As a consequence, we obtain necessary and sufficient conditions for which the m-fold symmetric positive projective tensor product ⊗̂m,s,|π|E (resp. the m-fold symmetric positive injective tensor product ⊗̌m,s,| |E) has a shrinking basis or a boundedly complete basis. For a vector lattice E and n ∈ N, let ⊗̄n,sE denote the n-fold Fremlin vector lattice symmetric tensor product of E. For m,n ∈ N with m > n, we prove that (i) if ⊗̄m,sE is uniformly complete then ⊗̄n,sE is positively isomorphic to a complemented subspace of ⊗̄m,sE, and (ii) if there exists φ ∈ E∼ + such that ker(φ) is a projection band in E then ⊗̄n,sE is lattice isomorphic to a projection band of ⊗̄m,sE. We also obtain analogous results for the n-fold Fremlin Banach lattice symmetric tensor product ⊗̂n,s,|π|E of E where E is a Banach lattice.