Abstract
We note that the well-known result of von Neumann (Contrib Theory Games 2:5–12, 1953) is not valid for all doubly substochastic operators on discrete Lebesgue spaces \(\ell ^p(I)\), \(p\in [1,\infty )\). This fact lead us to distinguish two classes of these operators. Precisely, the class of increasable doubly substochastic operators on \(\ell ^p(I)\) is isolated with the property that an analogue of the Von Neumann result on operators in this class is true. The submajorization relation \(\prec _s\) on the positive cone \(\ell ^p(I)^+\), when \(p\in [1,\infty )\), is introduced by increasable substochastic operators and it is provided that submajorization may be considered as a partial order. Two different shapes of linear preservers of submajorization \(\prec _s\) on \(\ell ^1(I)^+\) and on \(\ell ^p(I)^+\), when I is an infinite set, are presented.
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Acknowledgements
The authors are grateful to the reviewers for careful reading of the paper and valuable suggestions and comments. This research was financially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Contract No. 451-03-9/2021-14/200109) and by the bilateral project between Serbia and Slovenia (Generalized inverses, operator equations and applications, Grant No. 337-00-21/2020-09/32).
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Communicated by Catalin Badea.
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Ljubenović, M.Z., Rakić, D.S. Submajorization on \(\ell ^p(I)^+\) determined by increasable doubly substochastic operators and its linear preservers. Banach J. Math. Anal. 15, 60 (2021). https://doi.org/10.1007/s43037-021-00143-9
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DOI: https://doi.org/10.1007/s43037-021-00143-9