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The talk will consist of two parts. In the first part we develop the cornerstone the-orem given in [2, Proposition 2.1], which states that for a Banach lattice E with order contin-uous norm if D is a PL-compact subset of E then χ(D)=ρ(D), by showing that if a Banach lattice F has the positive Schur property then ρ(D)= ω(D) for any norm bounded subset D of F. Here, χ, ρ, and ω are Hausdorff measure of non-compactness, the measure of non-semicompactness introduced in [2], and the measure of weak non-compactness, respectively. Secondly, we in-troduce 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 re-sults obtained in [1] by showing that the modulus of non-semicompact convexity is continu-ous and has some extra properties in reflexive Banach lattices.
References:
[1] Y. Banas, On modulus of noncompact convexity and its properties. Canad. Math. Bull., 30 (1987), 186-192.
[2] B. de Pagter & A.R. Schep, Measures of non-compactness of operators on Banach lat-tices, J. Funct. Anal., 78 (1988), 31-55.