A note on the $A$-numerical range of semi-Hilbertian operators
Functional Analysis
2024-02-09 v1
Abstract
In this paper we explore the relation between the -numerical range and the -spectrum of -bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of -normal operator and prove that closure of the -numerical range of an -normal operator is the convex hull of the -spectrum. We further prove Anderson's theorem for the sum of -normal and -compact operators which improves and generalizes the existing result on Anderson's theorem for -compact operators. Finally we introduce strongly -numerically closed class of operators and along with other results prove that the class of -normal operators is strongly -numerically closed.
Cite
@article{arxiv.2402.05360,
title = {A note on the $A$-numerical range of semi-Hilbertian operators},
author = {Anirban Sen and Riddhick Birbonshi and Kallol Paul},
journal= {arXiv preprint arXiv:2402.05360},
year = {2024}
}