Skip to main content
SpringerLink
Log in

Riesz–Kantorovich formulas for operators on multi-wedged spaces

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz–Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Aliprantis, C.D., Tourky, R., Yannelis, N.C.: The Riesz–Kantorovich formula and general equilibrium theory. J. Math. Econ. 34(1), 55–76 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aliprantis, C.D., Tourky, R., Yannelis, N.C.: A theory of value with non-linear prices: equilibrium analysis beyond vector lattices. J. Econ. Theory 100(1), 22–72 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985)

    MATH  Google Scholar 

  4. Aliprantis, C.D., Tourky, R.: Cones and Duality. Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence, RI (2007)

    MATH  Google Scholar 

  5. Andô, T.: On fundamental properties of a Banach space with a cone. Pac. J. Math. 12, 1163–1169 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  6. de Jeu, M., Messerschmidt, M.: A strong open mapping theorem for surjections from cones onto Banach spaces. Adv. Math. 259, 43–66 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jameson, G.: Ordered Linear Spaces. Lecture Notes in Mathematics, vol. 141. Springer, Berlin (1970)

    Book  Google Scholar 

  8. Messerschmidt, M.: Geometric duality theory of cones in dual pairs of vector spaces. J. Funct. Anal. 269(7), 2018–2044 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the Claude Leon Foundation (first author) and by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) (both authors). Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.

Author information

Authors and Affiliations

  1. Unit for BMI, North-West University, Private Bag X6001, Potchefstroom, 2520, South Africa

    Christopher Schwanke & Marten Wortel

  2. Claude Leon Foundation, PO Box 15599, Vlaeberg, Cape Town, 8018, South Africa

    Christopher Schwanke

  3. DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Private Bag 3, Wits 2050, Gauteng, South Africa

    Christopher Schwanke & Marten Wortel

Authors
  1. Christopher Schwanke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marten Wortel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher Schwanke.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schwanke, C., Wortel, M. Riesz–Kantorovich formulas for operators on multi-wedged spaces. Positivity 22, 461–476 (2018). https://doi.org/10.1007/s11117-017-0521-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-017-0521-x

Keywords

Mathematics Subject Classification

Search

Navigation