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Some characterizations of Riesz spaces in the sense of strongly order bounded operators

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Abstract

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in \(E^{\sim \sim }\) and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if \(T_A\) is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded.

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Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments.

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Correspondence to Mohammad Bagher Farshbaf Moghimi.

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Jalili, S.A., Farshbaf Moghimi, M.B., Haghnejad Azar, K. et al. Some characterizations of Riesz spaces in the sense of strongly order bounded operators. Positivity 24, 117–127 (2020). https://doi.org/10.1007/s11117-019-00669-1

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Keywords

  • Strongly order bounded operator
  • b-Property
  • Order bounded
  • b-Order bounded
  • KB-space

Mathematics Subject Classification

  • 47B65
  • 46B40
  • 46B42