Note on the Structure of the Spaces of Matrix Monotone Functions
Note on the Structure of the Spaces of Matrix Monotone Functions
复制标题
关于矩阵单调函数空间结构的注记
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
J. Tomiyama
中科院分区：
文献类型：
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作者：
H. Osaka;J. Tomiyama
设n∈N和M n是n×n矩阵的代数。当f(A)≤f(B)对每一对自伴矩阵a，b∈Mn成立，使得a≤b和a，b的所有特征值都包含在I中时，我们称函数f(A)为n阶单调或n-单调。对于任意n个∈N和一个有限区间I，我们定义了Cn(I)类为环I上所有正实值连续函数f的集合，使得f(i∘)⊂(0，∞)，且对任一子集S⊂I∘都存在一个关于(0，∞)在S上的正Pick函数h，然后我们用一个算子不等式刻画了Cn([0，1)).此外，我们还证明了对每个nC2n([0，∞))⊊Pn+([0，∞)).
Let n ∈ N and M n be the algebra of n ×n matrices. We call a function f matrix monotone of order n or n-monotone in short whenever the inequality f(a) ≤ f(b) holds for every pair of selfadjoint matrices a, b ∈ M n such that a ≤ b and all eigenvalues of a and b are contained in I. The spaces for n-monotone functions is written as P n (I). For each n ∈ N and a finite interval I we define the class C n (I) by the set of all positive real-valued continuous functions f over I such that f(I∘) ⊂ (0, ∞) and for any subset S ⊂ I∘ there exists a positive Pick function h on (0, ∞) interpolating f on S. Then we characterize C n ([0, 1)) by an operator inequality. Moreover we show that for each nC2n([0, ∞)) ⊊ P n +([0, ∞)).