Browse > Article
http://dx.doi.org/10.4134/BKMS.2016.53.1.119

SOME OBSERVATIONS ON THE NUMERICAL INDEX AND THE POLYNOMIAL NUMERICAL INDEX  

Kim, Sun Kwang (Department of Mathematics Kyonggi University)
Publication Information
Bulletin of the Korean Mathematical Society / v.53, no.1, 2016 , pp. 119-126 More about this Journal
Abstract
In this paper, we study both the numerical index and the polynomial numerical index. First, we give a sufficient condition for a Banach space X to have lushness. Second, we study the relation between the renormings of a Banach space and the k-order polynomial numerical index. This shows that every real Banach spaces of dimension greater that 1 can be renormed to have 2-order polynomial numerical index ${\alpha}$ for any ${\alpha}{\in}[0,1/18)$.
Keywords
numerical index; polynomial numerical index; polynomial; renorming;
Citations & Related Records
연도 인용수 순위
  • Reference
1 A. Aviles, V. Kadets, M. Martin, J. Meri, and V. Shepelska, Slicely countably determined Banach spaces, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4871-4900.   DOI
2 F. F. Bonsall and J. Duncan, Numerical Ranges ii, London Math. Soc. Lecture Note Series 10, Cambridge 1973.
3 K. Boyko, V. Kadets, M. Martin, and J. Meri, Properties of lush spaces and applications to Banach spaces with numerical index 1, Studia Math. 190 (2009), no. 2, 117-133.   DOI
4 K. Boyko, V. Kadets, M. Martin, and D. Werner, Numerical index of Banach spaces and duality, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 93-102.   DOI
5 Y. S. Choi, D. Garcia, S. G. Kim, and M. Maestre, The polynomial numerical index of a Banach space, Proc. Edinb. Math. Soc. 49 (2006), no. 1, 39-52.
6 Y. S. Choi, D. Garcia, M. Maestre, and M. Martin, Polynomial numerical index for some complex vector-valued function spaces, Q. J. Math. 59 (2008), no. 4, 455-474.   DOI
7 E. Ed-dari and M. Khamsi, The numerical index of the $L_p$ space, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2019-2025.   DOI
8 C. Finet, M. Martin, and R. Paya, Numerical index and renorming, Proc. Amer. Math. Soc. 131 (2003), no. 3, 871-877.   DOI
9 D. Garcia, B. Grecu, M. Maestre, M. Martin, and J. Meri, Two-dimensional Banach spaces with polynomial numerical index zero, Linear Algebra Appl. 430 (2009), no. 8-9, 2488-2500.   DOI
10 N. Ghoussoub, G. Godefroy, B. Maurey, and W. Schachermayer, Some topological and geometrical structures in Banach spaces, Mem. Amer. Math. Soc. 70 (1987), no. 378, iv+116 pp.
11 L. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math. 93 (1971), 1005-1019.   DOI
12 V. Kadets, M. Martin, and R. Paya Recent progress and open questions on the numerical index of Banach spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 100 (2006), no. 1-2, 155-182.
13 J. Kim and H. J. Lee, Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices, J. Funct. Anal. 257 (2009), no. 4, 931-947.   DOI
14 S. G. Kim, M. Martin, and J. Meri, On the polynomial numerical index of the real spaces $c_0,\;{\ell}_1\;and\;{\ell}_{\infty} $, J. Math. Anal. Appl. 337 (2008), no. 1, 98-106.   DOI
15 H. J. Lee, Banach spaces with polynomial numerical index 1, Bull. London Math. Soc. 40 (2008), no. 2, 193-198.   DOI
16 H. J. Lee and M. Martin, Polynomial numerical indices of Banach spaces with 1-unconditional basis, preprint.
17 H. J. Lee, M. Martin, and J. Meri, Polynomial numerical indices of Banach spaces with absolute norm, Linear Algebra Appl. 435 (2011), no. 2, 400-408.   DOI
18 A. Lima, intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1-62.   DOI
19 G. Lopez, M. Martin, and R. Paya, Real Banach spaces with numerical index 1, Bull. London Math. Soc. 31 (1999), no. 2, 207-212.   DOI
20 A. Lima, intersection properties of balls in spaces of compact operators, Ann. inst. Fourier Grenoble 28 (1978), 35-65.   DOI
21 M. Martin, J. Meri, M. Popov, and B. Randrianantoanina, Numerical index of absolute sums of Banach spaces, J. Math. Anal. Appl. 375 (2011), no. 1, 207-222.   DOI