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The talk will consist of two parts. In the first part, we develop the cornerstone theorem given in [2, Proposition 2.1], which states that for a Banach lattice E with order continuous norm (OCN) if D is a PL-compact subset of E then χ(D)=ρ(D), by showing that if E has OCN, then w(D)≤ ρ(D); on the other hand, if E has the Schur property, then ρ(D)≤ w(D) for any norm bounded subset D of E. Here, χ, ρ, and w are Hausdorff measure of non-compactness, the measure of non-semicompactness introduced in [2], and the measure of weak non-compactness, respectively. Secondly, we introduce the notion of the modulus of non-semicompact convexity in Banach lattices defined with the help of the measure of non-semicompactness in Banach lattices. We extend the results obtained in [1] by showing that the modulus of non-semicompact convexity is continuous and has some extra properties in reflexive Banach lattices.

[1] Y. Banas, On modulus of noncompact convexity and its properties, Canad. Math. Bull. 30, 186-192, 1987.

[2] B. de Pagter & A.R. Schep, Measures of non-compactness of operators on Banach lattices, J. Funct. Anal. 78, 31-55, 1988.