Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces

S. G. Kim

doi:10.30970/ms.57.1.10-15

S. G. Kim Kyungpook National University

DOI: https://doi.org/10.30970/ms.57.1.10-15

Keywords: numerical radius points; numerical radius peak mappings.

Abstract

For $n\geq 2$ and a real Banach space $E,$ ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.
Let $$\Pi(E)=\Big\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}\Big\}.$$
For $T\in {\mathcal L}(^n E:E),$ we define $$\qopname\relax o{Nr}({T})=\Big\{[x^*, (x_1, \ldots, x_n)]\in \Pi(E): |x^{*}(T(x_1, \ldots, x_n))|=v(T)\Big\},$$
where $v(T)$ denotes the numerical radius of $T$.
$T$ is called {\em numerical radius peak mapping} if there is $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ such that $\qopname\relax o{Nr}({T})=\{\pm [x^{*}, (x_1, \ldots, x_n)]\}.$
In this paper, we investigate some class of numerical radius peak mappings in ${\mathcal
L}(^n l_p:l_p)$ for $1\leq p<\infty.$ Let $(a_{j})_{j\in \mathbb{N}}$ be a bounded sequence in $\mathbb{R}$ such that $\sup_{j\in \mathbb{N}}|a_j|>0.$
Define $T\in {\mathcal L}(^n l_p:l_p)$ by
$$T\Big(\sum_{i\in \mathbb{N}}x_i^{(1)}e_i, \cdots, \sum_{i\in \mathbb{N}}x_i^{(n)}e_i \Big)=
\sum_{j\in \mathbb{N}}a_{j}~x_{j}^{(1)}\cdots x_{j}^{(n)}~e_j.\qquad\eqno(*)$$
In particular is proved the following statements:\
$1.$\ If $1< p<+\infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0\in \mathbb{N}$ such that
$$|a_{j_0}|>|a_{j}|~\mbox{for every}~j\in \mathbb{N}\backslash\{j_0\}.$$

$2.$\ If $p=1$ then $T$ is not a numerical radius peak mapping in ${\mathcal L}(^n l_1:l_1).$

References

R.M. Aron, C. Finet and E. Werner, Some remarks on norm-attaining n-linear forms, Function spaces (Edwardsville, IL, 1994), 19–28, Lecture Notes in Pure and Appl. Math., 172, Dekker, New York, 1995.

E. Bishop, R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc., 67 (1961), 97–98.

Y.S. Choi, S.G. Kim, Norm or numerical radius attaining multilinear mappings and polynomials, J. London Math. Soc., 54 (1996), No2, 135–147.

S. Dineen, Complex analysis on infinite dimensional spaces, Springer-Verlag, London, 1999.

M. Jiménez Sevilla, R. Payá, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math., 127 (1998), 99–112.

Y.S. Choi, D. Garcia, S.G. Kim, M. Maestre, Norm or numerical radius attaining polynomial on C(K), J. Math. Anal. Appl., 295 (2004), 80–96.

S.G. Kim, The norming set of a bilinear form on $l_{infty}^2$, Comment. Math., 60 (2020), No1-2, 37–63.

S.G. Kim, The norming set of a polynomial in ${mathcal P}(^2 l_{infty}^2)$, Honam Math. J., 42 (2020), No3, 569–576.

S.G. Kim, The norming set of a symmetric bilinear form on the plane with the supremum norm, Mat. Stud., 55 (2021), No2, 171–180.

S.G. Kim, The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm, New Zealand J. Math., 51 (2021), 95–108.

S.G. Kim, Numerical radius points of {mathcal L}(^m l_{infty}^n: l_{infty}^n)$, to appear in New Zealand J. Math., 53 (2022).

S.G. Kim, Numerical radius points of a bilinear mapping in ${mathcal L}(^2 l_{1}^2: l_{1}^2)$, Preprint.