Infinite sequences of linear functionals, positive operator‐valued measures and Naimark extension theorem
Infinite sequences of linear functionals, positive operator‐valued measures and Naimark extension theorem
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线性泛函的无限序列、正算子值测度和奈马克扩展定理
DOI：
10.1112/blms/bdq005
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发表时间：
2010
影响因子：
0.9
通讯作者：
R. Beneduci
中科院分区：
文献类型：
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作者：
R. Beneduci
设F：ℬ(ℝ)→ℱ(ℋ)是实数的BorelHilbert-代数到σ空间ℋ上的有界正算子空间的交换正算子值测度。众所周知，存在一个自伴算子A和一个马尔可夫核μ(·)(λ)使得F(Δ)=μΔ(A)，其中Δ∈ℬ(ℝ)。我们证明了，对于任意Δ∈ℬ(ℝ)，如果F(Δ)是离散算子，则A是F的矩的线性组合。这一结果允许我们通过F的奈马克展开来刻画由A表示的量子能观量。这一结果与线性泛函序列的下列性质有关。考虑线性泛函的无限序列{Ti}i∈ℕ，使得Tif=∫f(T)dμt(I)，对应于Borelμ-代数∈ℕ([0，1])上的概率测度的无限序列{σ(·)(I)}iℬ，使得μ(·)(I)≠μ(·)(J)，其中i，J∈ℕ和I≠j.存在实有界的一对一连续函数f，使得Ti f=∫f(T) dμt(I) ≠ ∫f(T) dμt(J) = tjf，i， j ∊ ℕ， i≠j.
Let F : ℬ(ℝ)→ℱ(ℋ) be a commutative positive operator‐valued measure from the Borel σ‐algebra of the reals to the space of bounded positive operators on the Hilbert space ℋ. It is well known that there exist a self‐adjoint operator A and a Markov kernel μ(·)(λ) such that F(Δ) = μΔ(A), where Δ ∈ ℬ(ℝ). We prove that, for any Δ ∈ ℬ(ℝ), if F(Δ) is a discrete operator, then A is a linear combination of the moments of F. This result allows us to characterize the quantum observable represented by A by means of the Naimark dilation of F. The result is connected with the following properties of the sequences of linear functionals. Consider an infinite sequence of linear functionals {Ti}i ∈ ℕ, such that Tif = ∫ f(t) dμt(i), corresponding to an infinite sequence of probability measures {μ(·)(i)}i ∈ ℕ on the Borel σ‐algebra ℬ([0, 1]) such that μ(·)(i) ≠ μ(·)(j), where i, j ∈ ℕ and i≠j. There exists a real bounded one‐to‐one continuous function f such that Ti f=∫f(t) dμt(i) ≠ ∫f(t) dμt(j) = Tjf,i, j ∊ ℕ, i≠j.