A semigroup approach to the numerical range of operators on Banach spaces

TL;DR

This paper introduces the numerical spectrum for operators on Banach spaces, establishing its properties and relation to the numerical range, especially in the context of $C_0$-semigroups, highlighting its advantages over the traditional numerical range.

Contribution

It defines the numerical spectrum for unbounded operators on Banach spaces and explores its properties, including convexity, closure, and relation to the spectrum and numerical range.

Findings

Numerical spectrum is always closed, convex, and contains the spectrum.

For bounded operators on Hilbert spaces, numerical spectrum and numerical range coincide.

The approach emphasizes the connection to $C_0$-semigroup theory.

Abstract

We introduce the numerical spectrum $\sigma_n(A)\subset \mathbb{C}$ of an (unbounded) linear operator $A$ on a Banach space $X$ and study its properties. Our definition is closely related to the numerical range $W(A)$ of $A$ and always yields a superset of $W(A)$. In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, $\sigma_n(A)$ is always closed, convex and contains the spectrum of $A$. In the paper we strongly emphasise the connection of our approach to the theory of $C_0$-semigroups.