Lower bound inequalities for norms of symmetrized tensor powers (Q1044590)

Suppose that \(V\) is a complex inner product space of positive dimension \(m\) with inner product \(\langle.,.\rangle\), and let \({T}_{n}(V)\) denote the set of all \(n\)-linear complex-valued functions defined on \(V\times V\times \dots\times V\) (\(n\)-copies), the set of all symmetric members of \({T}_{n}(V)\) by \(S_{n}(V)\). The author extends the inner product \(\langle.,.\rangle\) on \(V\) to \({T}_{n}(V)\) in the usual way, and defines multiple tensor products \(A_{1}\otimes A_{2}\otimes\dots\otimes A_{n}\) and symmetric products \(A_{1}\cdot A_{2}\cdot\dots\cdot A_{n}\), where \(q_{1},q_{2} ,\dots,q_{n}\) are positive integers and \(A_{i}\in T_{q_{i}}(V)\) for each \(i\), as expected. If \(A\in S_{n}(V)\), then \(A^{k}\) denotes the symmetric product \(A\cdot A\cdot\dots\cdot A\) where there are \(k\) copies of \(A\). This paper is concerned with producing the best lower bounds for \(\|A^{k}\|^{2}\), particularly when \(n=2\). In this case, the author shows that \(\|A^{k}\|^{2}\) is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, \(M_{A}\), that is closely related to \(A\). From this, the author obtains many lower bounds for \(\|A^{k}\|^{2}\). In particular, it is shown that if \(a\) denotes \(1/r\) where \(r\) is the rank of \(M_{A}\), and \(A\neq0\), then \[ \|A^{k}\|^{2}\geq\frac{r(r+2)(r+4)\dots(r+2(k-1))} {r^{k}(2k-1)(2k-3)\dots3\cdot1}(\|A\|^{2})^{k}=\left[\prod_{t=0}^{k-1}\frac{(1+2at)}{(1+2t)}\right] (\|A\|^{2})^{k} \] for all integers \(k_{1}\), with equality in case \(k\geq2\) if and only if \(M_{A}\) is a nonnegative multiple of a Hermitian idempotent. A similar, but independent inequality is that \[ \|A^{k}\|^{2}\geq\lambda_{1}^{k}+\lambda_{2}^{k} +\dots+\lambda_{m}^{k}, \] where \(\lambda_{1},\lambda_{2},\dots,\lambda_{m}\) are the eigenvalues of \(M_{A}\).