The numerical ranges and the smooth points of the unit sphere. (English) Zbl 0697.47003

Let \(S_ p\) be the unit sphere of a complex Banach space (E,p) and \(F_ p\) the set of all smooth points on \(S_ p\). Assuming that the set \(F_ p\) is dense in \(S_ p\), the author proves that for continuous operators T the closure of the set \(\{p'(x,Tx)-ip'(x,iTx):\quad x\in F_ p\}\) is the closure of a Lumer numerical range of T, where \(p'\) denotes the Gâteaux derivative of p.
Reviewer: T.Nakazi


47A12 Numerical range, numerical radius