Three kinds of numerical indices of \(l_p\)-spaces

In this paper, we investigate the polynomial numerical index \(n^{(k)}(l_p),\) the symmetric multilinear numerical index \(n_s^{(k)}(l_p),\) and the multilinear numerical index \(n_m^{(k)}(l_p)\) of \(l_p\) spaces, for \(1\leq p\leq \infty.\) First we prove that \(n_{s}^{(k)}(l_1)=n_{m}^{(k)}(l_1)=1,\) for every \(k\geq 2.\) We show that for \(1 \lt p \lt \infty,\) \(n_I^{(k)}(l_p^{j+1})\leq n_I^{(k)}(l_p^j),\) for every \(j\in \mathbb{N}\) and \(n_I^{(k)}(l_p)=\lim_{j\to \infty}n_I^{(k)}(l_p^j),\) for every \(I=s, m,\) where \(l_p^j=(\mathbb{C}^j, \|\cdot\|_p)\) or \((\mathbb{R}^j, \|\cdot\|_p).\) We also show the following inequality between \( n_s^{(k)}(l_p^j)\) and \(n^{(k)}(l_p^j)\): let \(1 \lt p \lt \infty\) and \(k\in \mathbb{N}\) be fixed. Then \[ c(k: l_p^j)^{-1}~n^{(k)}(l_p^j)\leq n_s^{(k)}(l_p^j)\leq n^{(k)}(l_p^j), \] for every \(j\in \mathbb{N}\cup\{\infty\},\) where \(l_p^{\infty}:=l_p,\) \[ c(k: l_p)=\inf\Big\{M>0: \|\check{Q}\|\leq M\|Q\|,\mbox{ for every}~Q\in {\mathcal P}(^k l_p)\Big\} \] and \(\check{Q}\) denotes the symmetric \(k\)-linear form associated with \(Q.\) From this inequality, we deduce that if \(l_{p}\) is a complex space, then \(\lim_{j\to \infty} n_s^{(j)}(l_p)=\lim_{j\to \infty} n_m^{(j)}(l_p)=0,\) for every \(1\lt p \lt \infty.\)