Numerical radius of rank-1 operators on Banach spaces (Q2922410)

Let \(X\) be a Banach space, \(S_X\) be its unit sphere, \(L(X)\) be the space of all bounded linear operators \(T: X \to X\). The numerical radius of \(T \in L(X)\) is the following quantity: NEWLINE\[NEWLINE \nu(T) = \sup\{\left|x^*(Tx)\right|: x \in S_X, \, x^* \in S_{X^*},\, x^*(x) = 1\}. NEWLINE\]NEWLINE The numerical index of \(X\) is is defined as \(n(X) = \inf\{\nu(T): T \in S_{L(X)}\}\). According to the Bohnenblust-Karlin theorem, \(n(X) \geq 1/e\) for every complex space \(X\), but for real spaces this theorem is not valid: in \(\mathbb R^2\), equipped with the standard Euclidean norm, the \(\pi/2\)-rotation operator has numerical radius equal to 0.NEWLINENEWLINEThe authors introduce the rank-1 numerical index \(n_1(X)\) of a Banach space \(X\) as the infimum of the numerical radii of those \(T \in S_{L(X)}\) which have one-dimensional image. They demonstrate that \(n_1(X) \geq 1/e\) even for real spaces, and present an example of two-dimensional real space \(X\) with \(n_1(X) = 1/e\). The basic properties of this new index are studied, and some examples and open questions are discussed.