  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.
