Lieb Inequality And BMV Conjecture In Non-commutative Banach Spaces

This paper contains two mainly parts .In the first part, the purpose was to get the muliti-valued operator Lieb inequality.The properties of the singular value ofÏ„?measurable operator and the properties of themonotonic increasing convex function were used to generalize the conclusion aboutmatrix in [1] to the situation ofÏ„?measurable operator.Further extended to the caseof the muliti-valued operator, we got the result as follows: If A, B∈M₀ are positiveoperators, AB is normal and l∈ρ(AB) = C\σ(AB) ( l is simple line connecting originand∞).Then, we have(AB)^r<rB^r , for r≥1 ;A^rB^r<<(AB)^r , for 0≤r≤1 .In the second part, the BMV conjecture about matrix was generalized to the case ofÏ„-measurable operator. The proof was given in some cases. For example, the BMVconjecture is conforming when A, B are projections, i.e.Ï„((A + tB)^m) have nonnegativeTaylor coeffcients.