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Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 585–594 | Cite as

Norm Ratios Under a Weak Order Relation in \({\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}\)

  • Tsuyoshi Ando
Article
  • 44 Downloads

Abstract

In the real Hilbert space of self-adjoint elements of the tensor product \({\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}\), there are two natural cones besides the cone \({\mathfrak {P}}_{0}\) of positive semi-definite elements. The one is and the other is the cone \({\mathfrak {P}}_{-}\), dual to \({\mathfrak {P}}_{+}\) with respect to the inner product. Then, \({\mathfrak {P}}_{+} \subset {\mathfrak {P}}_{0} \subset {\mathfrak {P}}_{-}.\) A weak order relation ≽ is introduced by Our interest is in finding bounds for the ratio | | |T| | |/| | |S| | | for ST ≽ 0, where | | |⋅| | | is one of the operator norm, the trace norm, and the Hilbert-Schmidt norm.

Keywords

Norm ratios Cones of matrices Weak order relation Decompositions of matrices 

Mathematics Subject Classification (2010)

Primary 15B48 Secondary 15A60 15A69 

References

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    Alpirantis, Ch.D., Tourky, R.: Cone and Duality. American Mathematical Society, Providence (2007)Google Scholar
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    Ando, T.: Cones and norms in the tensor product of matrix spaces. Linear Algebra Appl. 379, 3–41 (2004)MathSciNetCrossRefGoogle Scholar
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    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefGoogle Scholar
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    Gurvits, L., Barnum, H.: Largest separable balls around the maximally mixed bipartite quantum state. arXiv:quant-ph/0204159v2 (2002)

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Hokkaido University (Emeritus)HokkaidoJapan

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