## 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

### MSC:

 47A12 Numerical range, numerical radius 45P05 Integral operators

### Keywords:

numerical range; numerical radius; algebraic spectrum