Positivity DOI 10.1007/s11117-015-0328-6 Positivity Domination problem on Banach lattices and almost weak compactness Hamadi Baklouti · Mohamed Hajji Received: 14 August 2014 / Accepted: 22 January 2015 © Springer Basel 2015 Abstract In this paper we give several new results concerning domination problem in the setting of positive operators between Banach lattices. Mainly, it is proved that every positive operator R on a Banach lattice E dominated by an almost weakly compact operator T satisfies that the R 2 is almost weakly compact. Domination by strictly singular operators is also considered. Moreover, we present some interesting connections between strictly singular, disjointly strictly singular and almost weakly compact operators. Keywords Positive operator · Banach lattice · Domination problem · Strictly singular operator Mathematics Subject Classification 47A50 · 47B65 · 46B42 1 Introduction Let E and F be two Banach lattices and consider two operators 0 ≤ R ≤ T : E → F . The inequalities here are in the sense that 0 ≤ Rx ≤ Tx in F, whenever x ∈ E is positive. An interesting problem is that of finding conditions under which properties of T , related to the norm topology, will be inherited by R, this is the so called domination problem. An important notion for this theory is the order continuity of the norm. We say that a Banach lattice E has order continuous norm if every increasing order-bounded sequence 0 ≤ x n ∈ E is norm convergent. A number of authors have been interested in the domination problem in various contexts and settings. For compact operators, the result of Dodds and Fermlin [7] asserts that, if E ′ (the dual space of E ) and F H. Baklouti (B ) · M. Hajji Sfax University, Sfax, Tunisia e-mail: h.baklouti@gmail.com