A note on "More Operator Versions of the Schwarz Inequality"
A note on "More Operator Versions of the Schwarz Inequality"
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关于“施瓦茨不等式的更多算子版本”的注释
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
R. Mathias
中科院分区：
文献类型：
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作者：
R. Mathias
证明了对于任意(n+1)-正(可能是非线性)映射Φ和任意有界线性算子Ai，i=1，？，n，我们有[Φ(Ai*Aj)]i，j=1*≥[Φ(Ai)*Φ(Aj)]i，j=1*，如果“(n+1)-正”被“n-正”替换为“n-正”，则该命题是假的.这解决了Bhatia和Davis提出的与Schwartz不等式有关的问题，该问题可被视为非对易方差-协方差不等式[2]。
It is shown that for any (n + 1)-positive (possibly non-linear) map Φ and any bounded linear operators Ai,i = 1,¨,n we have [Φ(Ai*Aj)]i,j = 1*≥[Φ(Ai)*Φ(Aj)]i,j = 1*, and that the statement is false if "(n + 1)-positive" is replaced by "n-positive". This resolves an issue raised by Bhatia and Davis in relation to a Schwartz inequality which can be regarded as a non-commutative variance-covariance inequality [2]