When each continuous operator is regular, II

The following theorem is essentially due to L.~Kantorovich and B. Vulikh and it describes one of the most important classes of Banach lattices between which each continuous operator is regular. {\bf Theorem 1.1.} {\sl Let $E$ be an arbitrary L-space and $F$ be an arbitrary Banach lattice with Levi norm. Then ${\cal L}(E,F)={\cal L}^r(E,F),\ (\star)$ that is, every continuous operator from $E$ to $F$ is regular.} In spite of the importance of this theorem it has not yet been determined to what extent the Levi condition is essential for the validity of equality $(\star)$. Our main aim in this work is to prove a converse to this theorem by showing that for a Dedekind complete $F$ the Levi condition is necessary for the validity of $(\star)$. As a sample of other results we mention the following. {\bf Theorem~3.6.} {\sl For a Banach lattice $F$ the following are equivalent: {\rm (a)} $F$ is Dedekind complete; {\rm (b)} For all Banach lattices $E$, the space ${\cal L}^r(E,F)$ is a Dedekind complete vector lattice; {\rm (c)} For all L-spaces $E$, the space ${\cal L}^r(E,F)$ is a vector lattice.

A Characterization of Compact-friendly Multiplication Operators

Answering in the affirmative a question posed in [Y.A.Abramovich, C.D.Aliprantis and O.Burkinshaw, Multiplication and compact-friendly operators, Positivity 1 (1997), 171--180], we prove that a positive multiplication operator on any $L_p$-space (resp. on a $C(\Omega)$-space) is compact-friendly if and only if the multiplier is constant on a set of positive measure (resp. on a non-empty open set). In the process of establishing this result, we also prove that any multiplication operator has a family of hyperinvariant bands -- a fact that does not seem to have appeared in the literature before. This provides useful information about the commutant of a multiplication operator.Comment: To appear in Indag. Math., 12 page

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.)

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