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 (γ_ϕ ‖ • ‖ _Y )-continuous, where γ_ϕ 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：

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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 =＆Integral; _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。