On the numerical index of the real two-dimensional L_p space

Abstract

We compute the numerical index of the two-dimensional real $L_p$ space for $\frac{6}{5}⩽p⩽1+{\alpha }_0$ and ${\alpha }_1⩽p⩽6$, where ${\alpha }_0$ is the root of $f\left(x\right)=1+x^{-2}-(x^{-\frac{1}{x}}+x^{\frac{1}{x}})$ and $\frac{1}{1+{\alpha }_0}+\frac{1}{{\alpha }_1}=1$. This, together with the previous results in Merí and Quero [On the numerical index of absolute symmetric norms on the plane. Linear Multilinear Algebra. 2021;69(5):971–979] and Monika and Zheng [The numerical index of ${\ell }_p^2$. Linear Multilinear Algebra. 2022;1–6. Published online DOI:10.1080/03081087.2022.2043818], gives the numerical index of the two-dimensional real $L_p$ space for $\frac{6}{5}⩽p⩽6$.

KEYWORDS:

• Numerical radius
• numerical index
• $L_p$-spaces

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 46B20
• 47A12

Acknowledgments

The authors thank the referees for carefully reading the manuscript and for their comments which led to an improved version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Research partially supported by projects PGC2018-093794-B-I00 (MCIU/AEI/ERDF, UE), PID2021-122126NB-C31 funded by Ministerio de Ciencia e Innovación/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, P20-00255 (Junta de Andalucía/ERDF, UE), A-FQM-484-UGR18 (Junta de Andalucía/ERDF, UE), and FQM-185 (Junta de Andalucía/ERDF, UE). The second author is also supported by the Ph.D. scholarship FPU18/03057 (MECD)