Glasnik Matematicki, Vol. 57, No. 1 (2022), 49-61. $$

THREE KINDS OF NUMERICAL INDICES OF $l_p$-SPACES

Sung Guen Kim

Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
e-mail:sgk317@knu.ac.kr

Abstract.   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 $1\le p\le \mathrm{\infty }.$ First we prove that $n_s^{(k)}(l_1)=n_m^{(k)}(l_1)=1,$ for every $k\ge 2.$ We show that for $1<p<\mathrm{\infty },$ $n_I^{(k)}(l_p^{j+1})\le n_I^{(k)}(l_p^j),$ for every $j\in \mathbb{N}$ and $n_I^{(k)}(l_p)=\underset{j\to \mathrm{\infty }}{lim}{n}_I^{(k)}(l_p^j),$ for every $I=s,m,$ where $l_p^j=({\mathbb{C}}^j,\Vert \cdot {\Vert }_p)$ or $({\mathbb{R}}^j,\Vert \cdot {\Vert }_p).$ We also show the following inequality between $n_s^{(k)}(l_p^j)$ and $n^{(k)}(l_p^j)$: let $1<p<\mathrm{\infty }$ and $k\in \mathbb{N}$ be fixed. Then $$c(k:l_p^j{)}^{-1}{n}^{(k)}(l_p^j)\le n_s^{(k)}(l_p^j)\le n^{(k)}(l_p^j),$$ for every $j\in \mathbb{N}\cup \{\mathrm{\infty }\},$ where $l_p^{\mathrm{\infty }}:=l_p,$ $$c(k:l_p)=inf\{M>0:\Vert \check{Q}\Vert \le M\Vert Q\Vert ,\text{ for every}Q\in \mathcal{P}{(}^k{l}_p)\}$$ 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 $\underset{j\to \mathrm{\infty }}{lim}{n}_s^{(j)}(l_p)=\underset{j\to \mathrm{\infty }}{lim}{n}_m^{(j)}(l_p)=0,$ for every $1<p<\mathrm{\infty }.$

2020 Mathematics Subject Classification.   46A22, 46G20

Key words and phrases.   The polynomial numerical index, the symmetric multilinear numerical index, the multilinear numerical index

https://doi.org/10.3336/gm.57.1.04

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