Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant

الأصل · EN

We present some operator inequalities for positive linear maps that generalize and improve the derived results in some recent years. For instant, if A and B are positive operators and m,m',M,M' are positive real numbers satisfying either one of the condition 0<m ≤ B ≤ m' <M' ≤ A ≤ M or 0<m ≤ A ≤ m' <M' ≤ B ≤ M, then align* Φᵖ (A ∇ ᵥ B+2 r Mm (A⁻¹∇ B⁻¹- &A⁻¹ B⁻¹)) & ≤ (K(h) 4²/ᵖ⁻¹ Kʳ₁ (√ h')) ᵖ Φᵖ (A ν B) align* and align* Φᵖ (A ∇ ᵥ B+2 r Mm (A⁻¹∇ B⁻¹-& A⁻¹ B⁻¹)) &≤ (K(h) 4²/ᵖ⁻¹ Kʳ₁(√ h')) ᵖ (Φ(A) ν Φ(B))ᵖ, align* where Φ is a positive unital linear map, 0 ≤ ν≤ 1, p ≥ 2, r={ν,1-ν}, h=M/m, h'=M'm', K(h)=(1+h)²4h and r₁={2r,1-2r}. We also obtain a reverse of the Ando inequality for positive linear maps via the Kantorovich constant.