Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces Related to Young's Result

Let H be a Hilbert space. In this paper we show among others that, if the selfadjoint operators A and B satisfy the condition $0$ $<$ m≤ A, B≤ M, for some constants $m,$ $M,$ then {align*} 0& ≤ {m}{M²}ν ( 1-ν ) ( {A²⊗ 1+1⊗ B²}{2}-A⊗ B) \\ & ≤ ( 1-ν ) A⊗ 1+ν 1⊗ B-A1-ν⊗ Bν \\ & ≤ {M}{m²}ν ( 1-ν ) ( {A²⊗ 1+1⊗ B²}{2}-A⊗ B) {align*} for all ν ∈ [ 0,1] . We also have the inequalities for Hadamard product {align*} 0& ≤ {m}{M²}ν ( 1-ν ) ( {A²+B²}{2}% ∘ 1-A∘ B) \\ & ≤ [ ( 1-ν ) A+ν B] ∘ 1-A1-ν∘ Bν \\ & ≤ {M}{m²}ν ( 1-ν ) ( {A²+B²}{2}% ∘ 1-A∘ B) {align*} for all ν ∈ [ 0,1] .