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Abstract

Let $E$ be a Banach function space over a finite and atomless measure space $(\Omega,\Sigma,\mu)$ and let $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ be real Banach spaces. A linear operator $T$ acting from the Köthe-Bochner space $E(X)$ to $Y$ is said to be absolutely continuous if $\|T({\bf 1}_{A_n}f)\|_Y\rightarrow 0$ whenever $\mu(A_n)\rightarrow 0$, $(A_n)\subset\Sigma$. In this paper we examine absolutely continuous operators from $E(X)$ to $Y$. Moreover, we establish relationships between different classes of linear operators from $E(X)$ to $Y$.