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On the numerical index of Banach spaces. (English) Zbl 1073.47006
The numerical radius of a bounded linear operator $T$ on a real or complex Banach space $X$ is, by definition, the quantity $$v(T)= \sup\{|x^*(Tx)|: x\in X,\,\Vert x\Vert= 1,\,x^*\in X^*,\,\Vert x^*\Vert= 1,\,x^*(x)= 1\}.$$ The numerical index of $X$ is $n(X)= \text{inf}\{v(T): T\in B(X),\,\Vert T\Vert= 1\}$. It is known that $\{n(X): X\text{ is a real Banach space}\}=[0,1]$ and $\{n(X):X\text{ is a complex Banach space}\}=[1/e, 1]$. In this paper, the author proves three more results on $n(X)$ for $X$ the $\ell_p$ and $L_p$ spaces: (1) $n(\ell_p)= \lim_{n\to\infty}(\ell^n_p)$ for $1< p<\infty$, (2) $M_p/2\le n(\ell^2_p)\le M_p$, where $M_p= \sup_{t\in [0,1]}(t^{p-1}- t)/(1+ t^p)$, for $1\le p\le 2$ and the real space $\ell^2_p$, and (3) $n(L_p(\mu))\ge n(\ell_p)$ for $1\le p<\infty$ and any positive measure $\mu$.

47A12Numerical range and numerical radius of linear operators
15A60Applications of functional analysis to matrix theory
Full Text: DOI
[1] F.F. Bonsall, J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Note Ser. 2, Cambridge University Press, 1971 · Zbl 0207.44802
[2] F.F. Bonsall, J. Duncan, Numerical ranges II, London Math. Soc. Lecture Note Ser. 10, Cambridge University Press, 1971 · Zbl 0207.44802
[3] Bohnenblust, H. F.; Karlin, S.: Geometrical properties of the unit sphere in Banach algebra. Ann. math. 62, 217-229 (1955) · Zbl 0067.35002
[4] Duncan, J.; Mcgregor, C. M.; Pryce, J. D.; White, A. J.: The numerical index of a normed space. J. London math. Soc. 2, No. 2, 481-488 (1970) · Zbl 0197.10402
[5] Finet, C.; Martín, M.; Payá, R.: Numerical index and renorming. Proc. amer. Math. soc. 131, No. 3, 871-877 (2003) · Zbl 1016.46011
[6] Glickfeld, B. W.: On an inequality of Banach algebra geometry and semi-inner product space theory. Illinois J. Math. 14, 76-81 (1970) · Zbl 0189.13304
[7] Gustafsan, K. E.; Rao, D. K. M.: Numerical range. The field of values of linear operators and matrices. (1997)
[8] Lumer, G.: Semi-inner-product spaces. Trans. amer. Math. soc. 100, 29-43 (1961) · Zbl 0102.32701
[9] Martín, M.: A survey on the numerical index of Banach space. Extracta math. 15, 265-276 (2000) · Zbl 0980.46006
[10] Martín, M.; Payá, R.: Numerical index of vector-valued function spaces. Studia math. 142, No. 3, 269-280 (2000) · Zbl 0986.46022