scholarly journals A characterization of discrete Banach lattices with order continuous norms

1988 ◽  
Vol 104 (1) ◽  
pp. 197-197 ◽  
Author(s):  
Witold Wnuk
Keyword(s):  
Banach Lattices ◽  
Order Continuous

scholarly journals On the similarity to nonnegative and Metzler Hessenberg forms

2021 ◽  
Vol 10 (1) ◽  
pp. 1-8
Author(s):  
Christian Grussler ◽  
Anders Rantzer
Keyword(s):  
Standard Form ◽  
Complete Characterization ◽  
Nonnegative Matrices ◽  
Positive Systems ◽  
Third Order ◽  
Hessenberg Form ◽  
Hessenberg Matrices ◽  
Open Question ◽  
Order Continuous

Abstract We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n 4. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n 5 remain similar to Metzler Hessenberg matrices.

scholarly journals Cone characterization of reflexive Banach lattices

1995 ◽  
Vol 37 (1) ◽  
pp. 65-67 ◽  
Author(s):  
Ioannis A. Polyrakis
Keyword(s):  
Banach Lattice ◽  
Normal Cone ◽  
Banach Lattices

AbstractWe prove that a Banach lattice X is reflexive if and only if X+ does not contain a closed normal cone with an unbounded closed dentable base.

Interpolation of weighted Banach lattices. A characterization of relatively decomposable Banach lattices

2003 ◽  
Vol 165 (787) ◽  
pp. 0-0 ◽  
Author(s):  
Michael Cwikel ◽  
Per G. Nilsson ◽  
Gideon Schechtman
Keyword(s):  
Banach Lattices

Converses for the Dodds–Fremlin and Kalton–Saab theorems

1996 ◽  
Vol 120 (1) ◽  
pp. 175-179 ◽  
Author(s):  
A. W. Wickstead
Keyword(s):  
Positive Operator ◽  
Banach Lattices ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Full Generality ◽  
Order Continuous

The two theorems in the title give conditions on Banach lattices E and F under which a positive operator from E into F, dominated by another positive operator with some property, must also have that property. The Dodds-Fremlin theorem says that this is true for compactness provided both E′ and F have order continuous norms, whilst the Kalton–Saab theorem establishes such a result for Dunford–Pettis operators provided F has an order continuous norm. These results were originally provided, in their full generality, in [3] and [5], respectively, whilst very readable proofs may be found in chapter 5 of [2] or §3·7 of [6].

scholarly journals Convex semigroups on $$L^p$$-like spaces

2021 ◽  
Author(s):  
Robert Denk ◽  
Michael Kupper ◽  
Max Nendel
Keyword(s):  
Semigroup Theory ◽  
Banach Lattices ◽  
Heat Equations ◽  
Well Posedness ◽  
Strong Continuity ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Typical Application ◽  
Convex Case ◽  
Order Continuous

AbstractIn this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $$L^p$$ L p -spaces in mind as a typical application. We show that the basic results from linear $$C_0$$ C 0 -semigroup theory extend to the convex case. We prove that the generator of a convex $$C_0$$ C 0 -semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of $$C_0$$ C 0 -semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.

scholarly journals Geometric Characterization of Injective Banach Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 250
Author(s):  
Anatoly Kusraev ◽  
Semën Kutateladze
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Set Theory ◽  
Banach Lattice ◽  
Dual Space ◽  
Banach Lattices ◽  
Transfer Principle ◽  
Ordered Banach Spaces

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.

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