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

Pietschmann, Frank; Rhodius, Adolf

Acta Sci. Math. 53, No. 3-4, 377-379 (1989).

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

MSC:

47A12

Numerical range, numerical radius

Keywords:

smooth points; Lumer numerical range; Gâteaux derivative