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

We develope new lower bounds for the A-numerical radius of semiHilbertian 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 semiHilbertian space operators.