Lower bound inequalities for norms of symmetrized tensor powers

Let VV be a complex inner product space of positive dimension mm with inner product 〈·,·〉〈·,·〉, and let Tn(V)Tn(V) denote the set of all nn-linear complex-valued functions defined on V×V×⋯×VV×V×⋯×V (nn-copies). By Sn(V)Sn(V) we mean the set of all symmetric members of Tn(V)Tn(V). We extend the inner product, 〈·,·〉〈·,·〉, on VV to Tn(V)Tn(V) in the usual way, and we define multiple tensor products A1⊗A2⊗⋯⊗AnA1⊗A2⊗⋯⊗An and symmetric products A1·A2⋯AnA1·A2⋯An, where q1,q2,…,qnq1,q2,…,qn are positive integers and Ai∈Tqi(V)Ai∈Tqi(V) for each ii, as expected. If A∈Sn(V)A∈Sn(V), then AkAk denotes the symmetric product A·A⋯AA·A⋯A where there are kk copies of AA. We are concerned with producing the best lower bounds for ‖Ak‖2‖Ak‖2, particularly when n=2n=2. In this case we are able to show that ‖Ak‖2‖Ak‖2 is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, MAMA, that is closely related to AA. From this we are able to obtain many lower bounds for ‖Ak‖2‖Ak‖2. In particular, we are able to show that if ωω denotes 1/r1/r where rr is the rank of MAMA, 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⩾1k⩾1, with equality in case k⩾2k⩾2 if and only if MAMA is a non-negative multiple of a Hermitian idempotent. A similar, but independent inequality is that ‖Ak‖2⩾λ1k+λ2k+⋯+λmk, where λ1,λ2,…,λmλ1,λ2,…,λm are the eigenvalues of MAMA.