Über numerische Wertebereiche und Spektralwertabschätzungen. (German) Zbl 0575.47005

Let p be a seminorm on a complex vector space E, T an endomorphism on E, \(S_ p\) is the unit sphere in (E,p) and \(\forall x\in S_ p,\quad D_ p(x)=\{f\in E':\quad f(x)=1,\quad | f(y)| \leq p(y),\quad y\in E\}.\) Assume that for the map \(Q_ p: S_ p\to Power\) set of E’ \(\emptyset \neq Q_ p(x)\subseteq D_ p(x),\quad \forall x\in S_ p.\) The set \(V_{Q_ p}(T)=\{f(Tx):\quad f\in Q_ p(x),\quad x\in S_ p\}\) is called the numerical range of T relative to \(Q_ p\), and \(v_ p(T)=\sup \{| \lambda |:\quad \lambda \in V_{D_ p}(T)\}\) is called the numerical radius of T. \(\sigma\) (T) denotes the algebraic spectrum of T. The author is interested in the study of \(\overline{V_{D_ p}(T)}\) and shows, for instance, that for a continuous endomorphism T of a complete seminormed space (E,p), \(\sigma (T)\setminus \sigma (T|_{F_ p})\subseteq \overline{V_{D_ p}(T)}\) where \(T|_{F_ p}\) denotes the restriction of T to the null space \(F_ p\) of p. He then applies his results to operators on Hilbert spaces, and also to special integral operators.
Reviewer: M.S.Ramanujan


47A12 Numerical range, numerical radius
45P05 Integral operators