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

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.

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Correspondence to Tsuyoshi Ando.

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Ando, T. Norm Ratios Under a Weak Order Relation in \({\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}\). Acta Math Vietnam 43, 585–594 (2018). https://doi.org/10.1007/s40306-018-0260-5

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Keywords

  • Norm ratios
  • Cones of matrices
  • Weak order relation
  • Decompositions of matrices

Mathematics Subject Classification (2010)

  • Primary 15B48
  • Secondary 15A60
  • 15A69