Maximal operators with respect to the numerical range

TL;DR

This paper studies $ $-maximal operators, which have numerical ranges within a convex set, providing characterizations in terms of their resolvent sets and encompassing examples like symmetric and dissipative operators.

Contribution

It offers new characterizations of $ $-maximal operators using resolvent set properties, extending understanding of their structure and examples.

Findings

Characterizations of $ $-maximal operators via resolvent sets

Connections to symmetric, accretive, and dissipative operators

Insights into generators of strongly continuous semigroups

Abstract

Let $\mathfrak{n}$ be a nonempty, proper, convex subset of $\mathbb{C}$. The $\mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $\mathfrak{n}$ and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the $\mathfrak{n}$-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.