LAUSR

In the real Hilbert space of self-adjoint elements of the tensor product Mm⊗Mn${\mathbb {M}}_{m}\otimes {\mathbb {M}}_{n}$, there are two natural cones besides the cone P0${\mathfrak {P}}_{0}$ of positive semi-definite elements. The one is and the other is the cone P−${\mathfrak {P}}_{-}$, dual to P+${\mathfrak {P}}_{+}$ with respect to the inner product. Then, P+⊂P0⊂P−.${\mathfrak {P}}_{+} \subset {\mathfrak {P}}_{0} \subset {\mathfrak {P}}_{-}.$ A weak order relation ≽ is introduced by Our interest is in finding bounds for the ratio | | |T| | |/| | |S| | | for S ≽T ≽ 0, where | | |⋅| | | is one of the operator norm, the trace norm, and the Hilbert-Schmidt norm.