A note on the $A$-numerical range of semi-Hilbertian operators

Abstract

In this paper we explore the relation between the $A$-numerical range and the $A$-spectrum of $A$-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of $A$-normal operator and prove that closure of the $A$-numerical range of an $A$-normal operator is the convex hull of the $A$-spectrum. We further prove Anderson's theorem for the sum of $A$-normal and $A$-compact operators which improves and generalizes the existing result on Anderson's theorem for $A$-compact operators. Finally we introduce strongly $A$-numerically closed class of operators and along with other results prove that the class of $A$-normal operators is strongly $A$-numerically closed.