## 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