$A$-numerical radius : New inequalities and characterization of equalities

Abstract: We develope new lower bounds for the $A$-numerical radius of semi-Hilbertian space operators, and applying these bounds we obtain upper bounds for the $A$-numerical radius of the commutators of operators. The bounds obtained here improve on the existing ones. Further, we provide characterizations for the equality of the existing $A$-numerical radius inequalities of semi-Hilbertian space operators.