Abstract
It is shown that for any (n + 1)-positive (possibly non-linear) map Φ and any bounded linear operators A i ,i = 1,¨,n we have [Φ(A i * A j )] i,j = 1 *≥[Φ(A i )*Φ(A j )] 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]
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Mathias, R. A note on "More Operator Versions of the Schwarz Inequality". Positivity 8, 85–87 (2004). https://doi.org/10.1023/B:POST.0000023200.14261.fc
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DOI: https://doi.org/10.1023/B:POST.0000023200.14261.fc