Abstract
Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let Lφ(X) denote the corresponding Orlicz-Bochner space and
$\begin{array}{}
\displaystyle
\mathcal T^\wedge_\varphi
\end{array}$ denote the finest Lebesgue topology on Lφ(X). We examine different classes of (
$\begin{array}{}
\displaystyle
\mathcal T^\wedge_\varphi
\end{array}$, ∥ ⋅ ∥Y)-continuous linear operators T : Lφ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.