Publication date: 30 June 2022
Volume: Vol 57
Issue: Svezak 1
Pages: 49-61
DOI: https://doi.org/10.3336/gm.57.1.04
Izvorni znanstveni članak
https://doi.org/10.3336/gm.57.1.04
Three kinds of numerical indices of l_p-spaces
Sung Guen Kim
; Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
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 \(1\lt p \lt \infty.\)
279799
28.6.2022.
Posjeta: 388 *