Communications in Mathematical Physics

, Volume 81, Issue 1, pp 89–96 | Cite as

An inequality for Hilbert-Schmidt norm

  • Huzihiro Araki
  • Shigeru Yamagami


For the absolute value |C|=(C*C)1/2 and the Hilbert-Schmidt norm ∥CHS=(trC*C)1/2 of an operatorC, the following inequality is proved for any bounded linear operatorsA andB on a Hilbert space
$$|| |A|---|B| ||_{HS} \leqq 2^{1/2} ||A - B||_{HS} $$
. The corresponding inequality for two normal state ϕ and ψ of a von Neumann algebraM is also proved in the following form:
$$d(\varphi ,\psi ) \leqq ||\xi (\varphi ) - \xi (\psi )|| \leqq 2^{1/2} d(\varphi ,\psi )$$
. Here ξ(χ) denotes the unique vector representative of a state χ in a natural positive coneP forM, andd(ϕ, ψ) denotes the Bures distance defined as the infimum (which is also the minimum) of the distance of vector representatives of ϕ and ψ. In particular,
$$||\xi (\varphi _1 ) - \xi (\varphi _2 )|| \leqq 2^{1/2} ||\xi _1 - \xi _2 ||$$
for any vector representatives ξ j of ϕ j ,j=1, 2.


Neural Network Statistical Physic Hilbert Space Normal State Complex System 
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Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Huzihiro Araki
    • 1
  • Shigeru Yamagami
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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