scholarly journals A note on the positive Schur property

1989 ◽  
Vol 31 (2) ◽  
pp. 169-172 ◽  
Author(s):  
Witold Wnuk
Keyword(s):  
Positive Operators ◽  
Banach Lattices ◽  
Schur Property ◽  
The Difference

The purpose of this note is to characterize those Banach lattices (E, ∥·∥) which have the property:an operator T: E → c0 is a Dunford-Pettis operator if and only if T is regular (*)(i.e., T is the difference of two positive operators). Our characterization generalizes a theorem recently proved by Holub [6] and Gretsky and Ostroy [4], who have remarked that the space L1[0, 1] has the property (*). The main result presented here is the following theorem.

Separable Banach lattices on which every bounded linear operator is regular

2011 ◽  
Vol 150 (3) ◽  
pp. 557-560
Author(s):  
A. W. WICKSTEAD
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Linear ◽  
The Difference

AbstractWe give a complete description of those separable Banach lattices E with the property that every bounded linear from E into itself is the difference of two positive operators.

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

2007 ◽  
Vol 59 (3) ◽  
pp. 614-637 ◽  
Author(s):  
C. C. A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Nuclear Operators ◽  
Linear Maps

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

scholarly journals Compact groups of positive operators on Banach lattices

2014 ◽  
Vol 25 (2) ◽  
pp. 186-205 ◽  
Author(s):  
Marcel de Jeu ◽  
Marten Wortel
Keyword(s):  
Positive Operators ◽  
Banach Lattices ◽  
Compact Groups

scholarly journals Spectrum and convergence of eventually positive operator semigroups

2021 ◽  
Author(s):  
Sahiba Arora ◽  
Jochen Glück
Keyword(s):  
Differential Equations ◽  
Positive Operator ◽  
Positive Operators ◽  
Banach Lattices ◽  
Time Behaviour ◽  
Major Step ◽  
Operator Semigroups ◽  
Positive Semigroups ◽  
Long Time Behaviour ◽  
Long Time

AbstractAn intriguing feature of positive $$C_0$$ C 0 -semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. In particular, the convergence of semigroup operators (strongly or in the operator norm) as time tends to infinity can be characterized by a set of simple spectral and compactness conditions. In the present paper, we show that similar theorems remain true for the larger class of (uniformly) eventually positive semigroups—which recently arose in the study of various concrete differential equations. A major step in one of our characterizations is to show a version of the famous Niiro–Sawashima theorem for eventually positive operators. Several proofs for positive operators and semigroups do not work in our setting any longer, necessitating different arguments and giving our approach a distinct flavour.

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