Lower bound inequalities for norms of symmetrized tensor powers

Abstract

Let V be a complex inner product space of positive dimension m with inner product 〈·,·〉, and let Tn(V) denote the set of all n-linear complex-valued functions defined on V×V×⋯×V (n-copies). By Sn(V) we mean the set of all symmetric members of Tn(V). We extend the inner product, 〈·,·〉, on V to Tn(V) in the usual way, and we define multiple tensor products A1⊗A2⊗⋯⊗An and symmetric products A1·A2⋯An, where q1,q2,…,qn are positive integers and Ai∈Tqi(V) for each i, as expected. If A∈Sn(V), then Ak denotes the symmetric product A·A⋯A where there are k copies of A. We are concerned with producing the best lower bounds for ‖Ak‖2, particularly when n=2. In this case we are able to show that ‖Ak‖2 is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, MA, that is closely related to A. From this we are able to obtain many lower bounds for ‖Ak‖2. In particular, we are able to show that if ω denotes 1/r where r is the rank of MA, and A≠0, then‖Ak‖2⩾r(r+2)(r+4)⋯(r+2(k-1))rk(2k-1)(2k-3)⋯3·1‖A‖2k=∏t=0k-1(1+2ωt)(1+2t)‖A‖2kfor all integers k⩾1, with equality in case k⩾2 if and only if MA is a non-negative multiple of a Hermitian idempotent. A similar, but independent inequality is that ‖Ak‖2⩾λ1k+λ2k+⋯+λmk, where λ1,λ2,…,λm are the eigenvalues of MA.