Some properties of the integration operators on the spaces F(p, q, s)

We study the closed range property and the strict singularity of integration operators acting on the spaces F(p, pα − 2, s). We completely characterize the closed range property of the Volterra companion operator I_g on F(p, pα − 2, s), which generalizes the existing results and answers a question raised in [A. Anderson, Integral Equations Operator Theory, 69 (2011), no. 1, 87–99]. For the Volterra operator J_g, we show that, for 0 < α ≤ 1, J_g never has a closed range on F (p, pα − 2, s). We then prove that the notions of compactness, weak compactness and strict singularity coincide in the case of J_g acting on F(p,p − 2, s).