, Volume 22, Issue 2, pp 461–476 | Cite as

Riesz–Kantorovich formulas for operators on multi-wedged spaces

  • Christopher Schwanke
  • Marten Wortel


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.


Multi-wedged spaces Multi-lattices Riesz–Kantorovich formulas 

Mathematics Subject Classification

06F20 46A40 



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.


  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press, Orlando (1985)zbMATHGoogle Scholar
  4. 4.
    Aliprantis, C.D., Tourky, R.: Cones and Duality. Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence, RI (2007)zbMATHGoogle Scholar
  5. 5.
    Andô, T.: On fundamental properties of a Banach space with a cone. Pac. J. Math. 12, 1163–1169 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Jeu, M., Messerschmidt, M.: A strong open mapping theorem for surjections from cones onto Banach spaces. Adv. Math. 259, 43–66 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jameson, G.: Ordered Linear Spaces. Lecture Notes in Mathematics, vol. 141. Springer, Berlin (1970)CrossRefGoogle Scholar
  8. 8.
    Messerschmidt, M.: Geometric duality theory of cones in dual pairs of vector spaces. J. Funct. Anal. 269(7), 2018–2044 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Unit for BMINorth-West UniversityPotchefstroomSouth Africa
  2. 2.Claude Leon FoundationVlaeberg, Cape TownSouth Africa
  3. 3.DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)GautengSouth Africa

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