Abstract
We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which does not have an almost isometric copy of \(\ell _1^2\), such that the second polarization constant for the numerical radius is maximal. We also give a negative answer for complex Banach spaces to the question of Choi and Kim (J Lond Math Soc (2) 54(1):135–147, 1996) whether \(\frac{\sum _{k=1}^m k^m ~\left( {\begin{array}{c}m\\ k\end{array}}\right) }{m!}\) is the optimal upper bound for the mth polarization constant for arbitrary \(m\in \mathbb {N}\). Finally, we generalize the result of García et al. (Proc Am Math Soc 142(4):1229–1235, 2014) that, for 2-homogeneous polynomials, the numerical index of a complex lush space is greater or equal than 1/3.
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Javier Falcó, Domingo García, and Manuel Maestre were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. Domingo García and Manuel Maestre were also supported by Prometeo PROMETEO/2017/102. Sun Kwang Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [NRF-2017R1C1B1002928] and [NRF-2020R1C1C1A01012267]. Han Ju Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [NRF-2020R1A2C1A01010377].
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Falcó, J., García, D., Kim, S.K. et al. Polarization Constant for the Numerical Radius. Mediterr. J. Math. 17, 153 (2020). https://doi.org/10.1007/s00009-020-01597-1
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DOI: https://doi.org/10.1007/s00009-020-01597-1