Tensor products, symmetric products, and permanents of positive semi-definite Hermitian matrices

Abstract

Abstract Let S k(C n) denote the symmetric complex valued k-linear functions on (C n)k. We prove the inequality where A,C ∈ S n (C m) and B,D ∈ S p(C m). Equality results if and only if there exist λ,u ∈ C such that A=λC and B= μD. If w 1,w 2,…,wk are in C m then G(w 1,w 2,…,wk ) denotes the Gram matrix associated with w 1,w 2,…,wk An application of the above inequality to decomposables yields the inequality where and