Measures under the flat norm as ordered normed vector space

Piotr Gwiazda, Anna Marciniak-Czochra, Horst Thieme

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

The space of real Borel measures (Formula presented.) on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone (Formula presented.) of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of (Formula presented.) are compact and semiflows on (Formula presented.) are continuous under much weaker conditions. In turn, the flat norm offers new challenges because (Formula presented.) is rarely complete and (Formula presented.) is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on (Formula presented.) and continuous semiflows. Both topics prepare for a dynamical systems theory on (Formula presented.).

Original languageEnglish (US)
Pages (from-to)1-34
Number of pages34
JournalPositivity
DOIs
StateAccepted/In press - May 25 2017

Fingerprint

Normed vector space
Ordered Vector Space
System theory
Vector spaces
Cones
Dynamical systems
Norm
Semiflow
Borel Measure
Systems Theory
Eigenvalue Problem
Metric space
Lipschitz
Cone
Dynamical system
Non-negative
Subset

Keywords

  • Dynamical systems
  • Homogeneous maps
  • Krein–Rutman theorem
  • Measures
  • Ordered normed vector space
  • Spectral radius

ASJC Scopus subject areas

  • Analysis
  • Theoretical Computer Science
  • Mathematics(all)

Cite this

Measures under the flat norm as ordered normed vector space. / Gwiazda, Piotr; Marciniak-Czochra, Anna; Thieme, Horst.

In: Positivity, 25.05.2017, p. 1-34.

Research output: Contribution to journalArticle

Gwiazda, Piotr ; Marciniak-Czochra, Anna ; Thieme, Horst. / Measures under the flat norm as ordered normed vector space. In: Positivity. 2017 ; pp. 1-34.
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