The numerical range of non-negative operators in Krein spaces

TL;DR

This paper introduces and characterizes the numerical and co-numerical ranges of non-negative operators in Krein spaces, revealing their relationship with the spectrum of the operator.

Contribution

It provides a new characterization of the numerical and co-numerical ranges for non-negative operators in Krein spaces, linking these ranges to the spectrum.

Findings

Non-zero spectrum is contained in the closure of the intersection of the numerical and co-numerical ranges.

Defined and characterized the Krein space numerical range and co-numerical range.

Established the relationship between the spectrum and the numerical ranges.

Abstract

We define and characterize the Krein space numerical range $W(A)$ and the Krein space co-numerical range $W_{\rm co}(A)$ of a non-negative operator $A$ in a Krein space. It is shown that the non-zero spectrum of $A$ is contained in the closure of $W(A)\cap W_{\rm co}(A)$.