On the $A$-$q$-Numerical Range of Operators in Semi-Hilbertian Spaces

TL;DR

This paper explores the properties of the $A$-$q$-numerical range of operators in semi-Hilbertian spaces, establishing fundamental results, bounds, and characterizations of $A$-nilpotent operators.

Contribution

It introduces new properties and bounds for the $A$-$q$-numerical range and characterizes the $A$-nilpotent operators' numerical range in semi-Hilbertian spaces.

Findings

Established spectral inclusion results for the $A$-$q$-numerical range.

Derived bounds for the $A$-$q$-numerical radius.

Showed the $A$-$q$-numerical range of an $A$-nilpotent operator with index 2 is a disk.

Abstract

This study investigates the $A$-$q$-numerical range of an operator within the framework of semi-Hilbertian spaces. Several fundamental properties of the $A$-$q$-numerical range are established, including spectral inclusion results and a disk union formula. Bounds for the $A$-$q$-numerical radius are derived, extending and generalizing previously known results. Finally, the notion of $A$-nilpotent operator is introduced, and it is shown that the $A$-$q$-numerical range of an $A$-nilpotent operator with index $2$ is a disk (open or closed) in the complex plane.