Sufficient conditions for normal structure of a Banach space are given. One of them implies reflexivity for Banach spaces with an unconditional basis, and also for Banach lattices.

We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L¹+L^{\infty }.$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L¹+L^{\infty }$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L¹+L^{\infty }$ is empty.

An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that b ⪯ a if and only if there is a doubly stochastic matrix A such that b = Aa. We prove that under monotonicity assumptions on the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that $(\widetilde{L^p},L^{\infty })$ is a Calderón couple for 1 ≤ p < ∞, where $\widetilde{L^p}$ is the Köthe dual of the Cesàro space $Ce{s}_{p^{\text{'}}}$ (or equivalently the down space $L_{\downarrow }^{p^{\text{'}}}$). In particular, $(\widetilde{L¹},L^{\infty })$ is a Calderón couple, which gives a...

We characterize Banach lattices under which each b-weakly compact (resp. b-AM-compact, strong type (B)) operator is L-weakly compact (resp. M-weakly compact).