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Integral representation of linear operators on Orlicz-Bochner spaces
Collectanea Mathematica ( IF 0.7 ) Pub Date : 2010 , DOI: 10.1007/bf03191233 Krzysztof Feledziak , Marian Nowak
Collectanea Mathematica ( IF 0.7 ) Pub Date : 2010 , DOI: 10.1007/bf03191233 Krzysztof Feledziak , Marian Nowak
Let (Ω, Σ, μ) be a σ-finite measure space and let\(\mathcal{L}(X,Y)\) stand for the space of all bounded linear operators between Banach spaces (X; ‖ • ‖
X
) and (Y; ‖ • ‖
Y
). We study the problem of integral representation of linear operators from an Orlicz-Bochner spaceL
ϕ(μ,X) toY with respect to operator measures\(m : \sum \to \mathcal{L}(X,Y) \). It is shown that a linear operatorT:L
ϕ (μ,X) →Y has the integral representationT(f = ∫Ω
f(ω)dm with respect to a ϕ*-variationally μ-continuous operator measurem if and only ifT is (γϕ ‖ • ‖
Y
)-continuous, where γϕ stands for a natural mixed topology onL
ϕ (μ,X). As an application, we derive Vitali-Hahn-Saks type theorems for families of operator measures.
中文翻译:
Orlicz-Bochner空间上线性算子的积分表示
让(Ω,Σ,μ)是σ-有限测度空间和让\(\ mathcal {L}(X,Y)\)代表所有有界线性Banach空间之间运营商(的空间X ;‖•‖ X)和(Ÿ ;‖•‖ ÿ)。我们研究了相对于算子测度\(m:\ sum \ to \ mathcal {L}(X,Y)\)从Orlicz-Bochner空间L ϕ(μ,X)到Y的线性算子的积分表示问题。。证明线性算子T:L ϕ(μ,X)→ Y具有积分表示T(f =∫ Ω ˚F(ω)DM相对于一φ * -variationallyμ-连续算量度米当且仅当Ť是(γ φ ‖•‖ Ý) -连续,其中γ φ代表在自然混合拓扑大号ϕ(μ,X)。作为应用,我们推导出了针对操作员度量系列的Vitali-Hahn-Saks型定理。
更新日期:2020-09-21
中文翻译:
Orlicz-Bochner空间上线性算子的积分表示
让(Ω,Σ,μ)是σ-有限测度空间和让\(\ mathcal {L}(X,Y)\)代表所有有界线性Banach空间之间运营商(的空间X ;‖•‖ X)和(Ÿ ;‖•‖ ÿ)。我们研究了相对于算子测度\(m:\ sum \ to \ mathcal {L}(X,Y)\)从Orlicz-Bochner空间L ϕ(μ,X)到Y的线性算子的积分表示问题。。证明线性算子T:L ϕ(μ,X)→ Y具有积分表示T(f =∫ Ω ˚F(ω)DM相对于一φ * -variationallyμ-连续算量度米当且仅当Ť是(γ φ ‖•‖ Ý) -连续,其中γ φ代表在自然混合拓扑大号ϕ(μ,X)。作为应用,我们推导出了针对操作员度量系列的Vitali-Hahn-Saks型定理。