turkmath.org

The talk will consist of two parts. In the rst part, we develope 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 $\chi (D) = \rho (D)$, by showing that if a Banach lattice $E$ has $OCN$, then $\omega (D) = (D)$ for every bounded subset $D$ of $E$. Here ,$\chi, \rho$ and $\omega$ are the Hausdor 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 dened with the help of the measure of non-semicompactness in Banach lattices. We extend the results obtained in [1] showing that the modulus of non-semicompact convexity is continuous and has some extra properties in re exive Banach lattices.