Inequalities on weighted classical pythagorean means, Tracy-Singh products, and Khatri-Rao products for hermitian operators

Abstract

We establish a number of operator inequalities between three kinds of means, namely, weighted arithmetic/harmonic/geometric means, and two kinds of operator products, namely, Tracy-Singh products and Khatri-Rao products. These results are valid under certain assumptions relying on (opposite) synchronization, comparability, and spectra of operators. Our results include tensor product of operators, and Tracy-Singh/Khatri-Rao products of matrices as special cases.