Banach lattices of L-weakly and M-weakly compact operators

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space.Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [2219]; Research Foundation of Namik Kemal UniversityNamik Kemal University [NKUBAP.00.10.AR.15.04]The first author was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) within the context of 2219-Post Doctoral Fellowship Program and by the Research Foundation of Namik Kemal University (Project No. NKUBAP.00.10.AR.15.04.)

Plasmodium P-Type Cyclin CYC3 Modulates Endomitotic Growth during Oocyst Development in Mosquitoes

Cell-cycle progression and cell division in eukaryotes are governed in part by the cyclin family and their regulation of cyclin-dependent kinases (CDKs). Cyclins are very well characterised in model systems such as yeast and human cells, but surprisingly little is known about their number and role in Plasmodium, the unicellular protozoan parasite that causes malaria. Malaria parasite cell division and proliferation differs from that of many eukaryotes. During its life cycle it undergoes two types of mitosis: endomitosis in asexual stages and an extremely rapid mitotic process during male gametogenesis. Both schizogony (producing merozoites) in host liver and red blood cells, and sporogony (producing sporozoites) in the mosquito vector, are endomitotic with repeated nuclear replication, without chromosome condensation, before cell division. The role of specific cyclins during Plasmodium cell proliferation was unknown. We show here that the Plasmodium genome contains only three cyclin genes, representing an unusual repertoire of cyclin classes. Expression and reverse genetic analyses of the single Plant (P)-type cyclin, CYC3, in the rodent malaria parasite, Plasmodium berghei, revealed a cytoplasmic and nuclear location of the GFP-tagged protein throughout the lifecycle. Deletion of cyc3 resulted in defects in size, number and growth of oocysts, with abnormalities in budding and sporozoite formation. Furthermore, global transcript analysis of the cyc3-deleted and wild type parasites at gametocyte and ookinete stages identified differentially expressed genes required for signalling, invasion and oocyst development. Collectively these data suggest that cyc3 modulates oocyst endomitotic development in Plasmodium berghei

Norms of basic elementary operators on algebras of regular operators

We show that if E is an atomic Banach lattice with an ordercontinuous norm, A, B ∈ Lr(E) and MA,B is the operator on Lr(E) defined by MA,B(T) = AT B then ||MA,B||r = ||A||r||B||r but that there is no real α &gt; 0 such that ||MA,B || ≥ α ||A||r||B ||r.</i