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.

中文翻译：

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正希尔伯特空间算子的泛函表示

在本文中，我们考虑了正希尔伯特空间算子在所讨论的希尔伯特空间的单位球面上定义的非负实函数结构上的一些忠实表示。这些表示将正算子之间的顺序关系转变为实函数之间的顺序关系。其中两个将运算符之间通常的 Löwner 顺序转换为函数之间的逐点顺序，另外两个将运算符之间的谱顺序转换为函数之间相同的逐点顺序。我们研究这些表示保留了哪些代数运算，因此表示函数具有哪种代数结构。我们研究了同一个正算子的不同表示函数之间的差异。最后，