On spatial numerical ranges of operators on Banach spaces (Q1098077)

The spatial numerical range V(T) of an operator T on a Banach space X is defined by \(V(T)=\{f(Tx):\) \(x\in X\), \(f\in X^*\), \(\| f\| =\| x\| =f(x)=1\}\). An operator T is called convexoid if \(\overline{V(T)}=ch(\sigma (T))\), where ch means convex hull and \(\sigma\) (T) the spectrum of T. Let E denote the set of all extreme points of \(\overline{V(T)}\) of a convexoid operator T. Concerning these concepts the author proves the following interesting results. Theorem 1. For a normal operator T on a smooth reflexive Banach space X, the spatial numerical range V(T) of T is closed and convex if, and only if \(E\cap \sigma_ c(T)\) is empty, where \(\sigma_ c(T)\) is the continuous spectrum of T. Theorem 2. For a convexoid operator T on a Banach space X, the following statements hold. (a) \(E\subset \sigma_{\pi}(T)\), (b) If \(E\subset \sigma_ p(T)\), then V(T) is closed and convex, where \(\sigma_ p(T)\), \(\sigma_{\pi}(T)\) denote the point and approximate point spectrum of T. Theorem 3. For a normal and isoabelian operator T on a smooth reflexive Banach space X, the spatial numerical range V(T) is closed if and only if \(\sigma (T)=\sigma_ p(T).\) The article contains also some more results concerning the spatial numerical range of convexoid and isoabelian operators.