On different definitions of numerical range

TL;DR

This paper explores the relationships between various definitions of numerical ranges in functional analysis, establishing conditions under which they coincide or differ, with implications for understanding operator behavior.

Contribution

It proves that the intrinsic numerical range equals the convex hull of the approximated spatial numerical range and identifies conditions for their equality.

Findings

Intrinsic numerical range equals convex hull of approximated spatial numerical range

Conditions established for the equality of approximated and spatial numerical ranges

Provides insights into the structure of numerical ranges in operator theory

Abstract

We study the relation between the intrinsic and the spatial numerical ranges with the recently introduced "approximated" spatial numerical range. As main result, we show that the intrinsic numerical range always coincides with the convex hull of the approximated spatial numerical range. Besides, we show sufficient conditions and necessary conditions to assure that the approximated spatial numerical range coincides with the closure of the spatial numerical range.