Generalization of the Lieb–Thirring–Araki Inequality and Its Applications

The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem.

On some Hölder type trace inequalities for operator weighted geometric mean

We obtain some Hölder type trace inequalities for operator weighted geometric mean. Some vector inequalities are also given.

Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces

D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.