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On the modulus of measures with values in topological Riesz spaces.

Lech Drewnowski, Witold Wnuk (2002)

Revista Matemática Complutense

The paper is devoted to a study of some aspects of the theory of (topological) Riesz space valued measures. The main topics considered are the following. First, the problem of existence (and, particularly, the so-called proper existence) of the modulus of an order bounded measure, and its relation to a similar problem for the induced integral operator. Second, the question of how properties of such a measure like countable additivity, exhaustivity or so-called absolute exhaustivity, or the properties...

Once more on positive commutators

Roman Drnovšek (2012)

Studia Mathematica

Let A and B be bounded operators on a Banach lattice E such that the commutator C = AB - BA and the product BA are positive operators. If the product AB is a power-compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. This answers an open question posed by Bračič et al. [Positivity 14 (2010)], where the study of positive commutators of positive operators was initiated.

Operator ideal properties of vector measures with finite variation

Susumu Okada, Werner J. Ricker, Luis Rodríguez-Piazza (2011)

Studia Mathematica

Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the...

Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure

Zbigniew Lipecki (2015)

Commentationes Mathematicae Universitatis Carolinae

Let 𝔐 and be algebras of subsets of a set Ω with 𝔐 , and denote by E ( μ ) the set of all quasi-measure extensions of a given quasi-measure μ on 𝔐 to . We give some criteria for order boundedness of E ( μ ) in b a ( ) , in the general case as well as for atomic μ . Order boundedness implies weak compactness of E ( μ ) . We show that the converse implication holds under some assumptions on 𝔐 , and μ or μ alone, but not in general.

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