Abstract
In this paper we consider some faithful representations of positive Hilbert space operators on structures of nonnegative real functions defined on the unit sphere of the Hilbert space in question. Those representations turn order relations between positive operators to order relations between real functions. Two of them turn the usual Löwner order between operators to the pointwise order between functions, another two turn the spectral order between operators to the same, pointwise order between functions. We investigate which algebraic operations those representations preserve, hence which kind of algebraic structure the representing functions have. We study the differences among the different representing functions of the same positive operator. Finally, we introduce a new complete metric (which corresponds naturally to two of those representations) on the set of all invertible positive operators and formulate a conjecture concerning the corresponding isometry group.
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The authors are grateful to the reviewer for the careful reading of the paper and discovering a small flaw in the original proof of Proposition 11.
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The first author acknowledges supports from the Ministry for Innovation and Technology, Hungary, Grant TUDFO/47138-1/2019-ITM and from the National Research, Development and Innovation Office of Hungary, NKFIH, Grant Nos. K115383, K134944. The second author was supported by the National Research, Development and Innovation Office of Hungary, NKFIH, Grant No. K134944.
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Molnár, L., Ramanantoanina, A. On Functional Representations of Positive Hilbert Space Operators. Integr. Equ. Oper. Theory 93, 2 (2021). https://doi.org/10.1007/s00020-020-02605-5
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DOI: https://doi.org/10.1007/s00020-020-02605-5