Operators on the space of vector-valued totally measurable functions

Let L ( X , Y ) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B ( Σ , X ) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖ T ( f n ) ‖ Y → 0 whenever a sequence of scalar functions ( ‖ f n ( ⋅ ) ‖ X ) is order convergent to 0 in B ( Σ ) . It is shown that a bounded linear operator T : B ( Σ , X ) → Y is σ-smooth if and only if its representing measure m : Σ → L ( X , Y ) is variationally semi-regular, i.e., m ˜ ( A n ) → 0 as A n ↓ ∅ (here m ˜ ( A ) stands for the semivariation of m on A ∈ Σ ). As an application, we show that the space L σ s ( B ( Σ , X ) , Y ) of all σ-smooth operators from B ( Σ , X ) to Y provided with the strong operator topology is sequentially complete. We derive a Banach–Steinhaus type theorem for σ-smooth operators from B ( Σ , X ) to Y. Moreover, we characterize countable additivity of measures m : Σ → L ( X , Y ) in terms of continuity of the corresponding operators T : B ( Σ , X ) → Y .