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A Viewpoint to Measure of Non-Compactness of Operators in Banach Spaces
Acta Mathematica Scientia ( IF 1 ) Pub Date : 2020-05-01 , DOI: 10.1007/s10473-020-0301-8
Qinrui Shen

This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection $$\mathcal{C}(X)$$ (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 $$\mathcal{C}(X)$$ onto $$\mathcal{F}(\Omega)$$ is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and $$\mathcal{F}(\Omega)$$ the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both $$E_\mathcal{C}=\overline{J\mathcal{C}-J\mathcal{C}}$$ and $$E_\mathcal{K}=\overline{J\mathcal{K}-J\mathcal{K}}$$ are Banach lattices and $$E_\mathcal{K}$$ is a lattice ideal of $$E_\mathcal{C}$$. The quotient space $$E_\mathcal{C}/E_\mathcal{K}$$ is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and $$(FQJ)\mathcal{C}$$ which is a closed cone is contained in the positive cone of C(K), where $$Q:E_\mathcal{C}\rightarrow{E_\mathcal{C}/E_\mathcal{K}}$$ is the quotient mapping and $$F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}$$ 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, then $$\mu(T)=\mu(T(B_X))=\parallel(FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$

中文翻译:

Banach空间中算子非紧性测度的一个观点

本文致力于处理 Banach 空间中算子的非紧性度量。首先,集合$$\mathcal{C}(X)$$(由赋有uaual集合加法和标度乘法的Banach空间X的所有非空闭有界凸集组成)是一个赋范半群,映射J来自$$\mathcal{C}(X)$$ 到 $$\mathcal{F}(\Omega)$$ 是一个完全保序的正线性满射等距,其中 Ω 是 X* 和 $$ 的封闭单位球\mathcal{F}(\Omega)$$ X* 上的所有连续和 w*-下半连续次线性函数的集合,但仅限于 Ω。此外,$$E_\mathcal{C}=\overline{J\mathcal{C}-J\mathcal{C}}$$ 和 $$E_\mathcal{K}=\overline{J\mathcal{K} -J\mathcal{K}}$$ 是 Banach 格,$$E_\mathcal{K}$$ 是 $$E_\mathcal{C}$$ 的格理想。商空间 $$E_\mathcal{C}/E_\mathcal{K}$$ 是一个抽象的 M 空间,因此,对于一些紧凑的 Haudorspace K,阶数与 C(K) 的子格等距,并且 $$(FQJ) \mathcal{C}$$ 是一个闭锥体,它包含在 C(K) 的正锥体中,其中 $$Q:E_\mathcal{C}\rightarrow{E_\mathcal{C}/E_\mathcal{K }}$$ 是商映射,$$F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}$$ 是对应的阶等距。最后给出算子非紧性度量的表示:设BX为Banach空间X的封闭单位球,则$$\mu(T)=\mu(T(B_X))=\parallel( FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$ 并且 $$(FQJ)\mathcal{C}$$ 包含在 C(K) 的正圆锥中,其中 $$Q:E_\mathcal{C}\rightarrow{E_\mathcal{C} /E_\mathcal{K}}$$ 是商映射,$$F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}$$ 是对应的阶等距。最后给出算子非紧性度量的表示:设BX为Banach空间X的封闭单位球,则$$\mu(T)=\mu(T(B_X))=\parallel( FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$ 并且 $$(FQJ)\mathcal{C}$$ 包含在 C(K) 的正圆锥中,其中 $$Q:E_\mathcal{C}\rightarrow{E_\mathcal{C} /E_\mathcal{K}}$$ 是商映射,$$F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}$$ 是对应的阶等距。最后给出算子非紧性度量的表示:设BX为Banach空间X的封闭单位球,则$$\mu(T)=\mu(T(B_X))=\parallel( FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$
更新日期:2020-05-01
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