134 research outputs found
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 be an
arbitrary L-space and be an arbitrary Banach lattice with Levi norm. Then
that is, every continuous operator
from to 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
. Our main aim in this work is to prove a converse to this theorem by
showing that for a Dedekind complete the Levi condition is necessary for
the validity of .
As a sample of other results we mention the following. {\bf Theorem~3.6.}
{\sl For a Banach lattice the following are equivalent: {\rm (a)} is
Dedekind complete; {\rm (b)} For all Banach lattices , the space is a Dedekind complete vector lattice; {\rm (c)} For all L-spaces
, the space 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 -space (resp. on a -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 > 0 such that ||MA,B || ≥ α ||A||r||B ||r.</i
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