A survey on the numerical index of a Banach space (Q2708476)

Let \(X\) be a Banach space and denote by \(S_X\) the unit sphere of \(X\) and by \(L(X)\) the Banach algebra of all bounded linear operators on \(X\). In this paper the author considers the so called numerical index of the space \(X\), namely the constant \(n(X)\) defined by \(n(X)=\text{ inf}\{v(T)~ :~T\in S_{L(X)}\}\), where \(v(T)\) denotes the numerical radius of the operator \(T\), next, in section 2 he summarizes some known properties of the numerical index, and computes it for some concrete spaces. In section 3 the facts known about Babach spaces with numerical index \(1\) are exposed and in section 4 some geometrical properties of Banach spaces implying numerical index \(1\), and the relations between them are discussed. Finally, in section 5 some remarks and open problems are also presented.