Operator Inequalities and Related Norm Inequalities

We set the following three actual research objectives：(1)study of the lattice structure of the cone of positive semi-definite operators，(2)study of super-and lower-subadditivity related to operator concave/convex functions，(3)study of convexity properties related to the trace.Concerning the fobjective，we have cceeded in determining the condition for a pair of operators to admit the inimum in the fobjective，the fimum in the confobjective|the fecty|the condition for a pair of operators to admit the fimum in the confobjective|the following the condition for a pair of operators！Which is valid for any non-negative operator concave function f(T)on the half-line[0，∞)，any pair of positive semi-definite operators A，B and any unitarily invariant norm|·|[Reference 3]。Concerning the third objective，we have obtained various estimates of the traces of multiple products of two matrices[Refernece5]。Also we have succeeded in generalizing some known determinant and trace inequalities to majorization type inequalities for eigenvalues[Reference 7]。

We set the following three actual research objectives :(1) study of the lattice structure of the cone of positive semi-definite operators,(2) study of super- and lower-subadditivity related to operator concave/convex functions,(3) study of convexity properties related to the trace.Concerning the first objective, we have succeeded in determining the condition for a pair of operators to admit the infimum in the cone of positive semi-definite operators [Reference 2].Concerning the second objective, we have succeeded in establishing the following inequality|||f (A+B)|||【less than or equal】 |||f (A)+f (B)|||,which is valid for any non-negative operator concave function f (t) on the half-line [0, ∞), any pair of positive semi-definite operators A,B and any unitarily invariant norm |||・|||[Reference 3].Concerning the third objective, we have obtained various estimates of the traces of multiple products of two matrices [Refernece 5]. Also we have succeeded in generalizing some known determinant and trace inequalities to majorization type inequalities for eigenvalues [Reference 7].