Schur majorization inequalities for symmetrized sums with applications to tensor products (Q1863518)

The author shows that if \(w\prec y\) and \(x\prec z\) are four vectors in \(\mathbb{R}^n\), then a number of Schur majorizations hold between ``symmetrized'' vector functions of \(w,x,y\) and \(z\), e.g., \((w_i+x_j)_{i,j}\prec (y_i+z_j)_{i,j}\) where the left-hand expression means the vector of dimension \(n^2\) consisting of all sums \(w_i+x_j\) of the co-ordinates of \(w\) and \(x\), arranged in lexicographic order. Vector and matrix versions of Muirhead's theorem for scalar inequalities have been obtained. Applications to the obtained matrix inequalities are given for tensor products, e.g., if \(A,B\) and \(C\) are Hermitian and \(\lambda(A)\prec\lambda(B)\), then \(\lambda(A\otimes C)\prec\lambda(B\otimes C)\).