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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 $\mu$.

MSC:
47A12Numerical range and numerical radius of linear operators
15A60Applications of functional analysis to matrix theory
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References:
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