Numerical range with respect to a family of projections

TL;DR

This paper introduces a new concept called the numerical range of a bounded linear operator relative to a family of projections, connecting it to classical and generalized numerical ranges.

Contribution

It defines the numerical range with respect to a family of projections and explores its relationship to existing numerical range concepts.

Findings

Provides a precise definition of the new numerical range concept.

Establishes connections to classical, quadratic, and block numerical ranges.

Lays groundwork for further analysis of operator properties using this new framework.

Abstract

In this note we introduce the concept of the numerical range of a bounded linear operator with respect to a family of projections. We give a precise definition and elaborate on its connection to the classical numerical range as well as to generalisations such as the quadratic numerical range and block numerical range.