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

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.