Three kinds of numerical indices of \(l_p\)-spaces
نویسندگان
چکیده
In this paper, we investigate the polynomial numerical index \(n^{(k)}(l_p),\) symmetric multilinear \(n_s^{(k)}(l_p),\) and \(n_m^{(k)}(l_p)\) of \(l_p\) spaces, for \(1\leq p\leq \infty.\) First prove that \(n_{s}^{(k)}(l_1)=n_{m}^{(k)}(l_1)=1,\) every \(k\geq 2.\) We show \(1 \lt p \infty,\) \(n_I^{(k)}(l_p^{j+1})\leq n_I^{(k)}(l_p^j),\) \(j\in \mathbb{N}\) \(n_I^{(k)}(l_p)=\lim_{j\to \infty}n_I^{(k)}(l_p^j),\) \(I=s, m,\) where \(l_p^j=(\mathbb{C}^j, \|\cdot\|_p)\) or \((\mathbb{R}^j, \|\cdot\|_p).\) also following inequality between \( n_s^{(k)}(l_p^j)\) \(n^{(k)}(l_p^j)\): let \infty\) \(k\in be fixed. Then \[ c(k: l_p^j)^{-1}~n^{(k)}(l_p^j)\leq n_s^{(k)}(l_p^j)\leq n^{(k)}(l_p^j), \] \mathbb{N}\cup\{\infty\},\) \(l_p^{\infty}:=l_p,\) l_p)=\inf\Big\{M>0: \|\check{Q}\|\leq M\|Q\|,\mbox{ every}~Q\in {\mathcal P}(^k l_p)\Big\} \(\check{Q}\) denotes \(k\)-linear form associated with \(Q.\) From inequality, deduce if \(l_{p}\) is a complex space, then \(\lim_{j\to \infty} n_s^{(j)}(l_p)=\lim_{j\to n_m^{(j)}(l_p)=0,\) \(1\lt
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ژورنال
عنوان ژورنال: Glasnik Matematicki
سال: 2022
ISSN: ['1846-7989', '0017-095X']
DOI: https://doi.org/10.3336/gm.57.1.04