On the $A$-$q$-Numerical Range of Operators in Semi-Hilbertian Spaces
Jyoti Rani, Arnab Patra, Riddhick Birbonshi
TL;DR
This work extends the theory of the $q$-numerical range to semi-Hilbertian spaces by developing the $A$-weighted framework $W_{q,A}(T)$ and its radius $w_{q,A}(T)$. It derives a circular union representation and proves spectral inclusion properties, along with invariance under $A$-unitaries and stability under $A$-approximate unitary equivalence, linking the semi-Hilbertian range to the classical $q$-range via lifting to $\mathbf{R}(A^{1/2})$. For $A$-self-adjoint operators, it provides an explicit elliptic-disk characterization with foci at $q\lambda_1$ and $q\lambda_m$ and minor axis $\sqrt{1-|q|^2}(\lambda_1-\lambda_m)$. The paper also introduces $A$-nilpotent operators and establishes that their $A$-$q$-numerical range is a disk centered at the origin when the index is $2$, with sharp bounds on $w_{q,A}(T)$ and a zero $A$-spectral radius, extending classical nilpotent results to semi-Hilbertian settings. Collectively, the results offer a unified, generalized framework for the study of numerical ranges under $A$-weighted inner products, with connections to the Davis-Wielandt structure and potential applications in operator theory under constrained geometries.
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.