We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space $\mathcal{ H}$ and a non-zero positive bounded linear operator $A$ on $\mathcal{ H},$ we show with among other seminorm inequalities, if $S,T,X\in \mathcal{ B}_A(\mathcal{ H})$, i.e., if $A$-adjoint of $S,T,X$ exist then $$2\|S^{ \sharp_A}XT\|_A \leq \|SS^{ \sharp_A}X+XTT^{ \sharp_A}\|_A.$$ Further, we prove that if $T\in \mathcal{ B}_A(\mathcal{ H})$ then \begin{ eqnarray*} \frac{ 1}{ 4}\|T^{ \sharp_{ A}}T+TT^{ \sharp_{ A}}\|_A \leq \frac{ 1}{ 8}\bigg( \|T+T^{ \sharp_{ A}}\|_A^2+\|T-T^{ \sharp_{ A}}\|_A^2\bigg), ~~\textit{ and} \end{ eqnarray*} \begin{ eqnarray*} \frac{ 1}{ 8}\bigg( \|T+T^{ \sharp_{ A}}\|_A^2+\|T-T^{ \sharp_{ A}}\|_A^2\bigg) +\frac{ 1}{ 8}c_A^2\big(T+T^{ \sharp_{ A}}\big)+\frac{ 1}{ 8}c_A^2\big(T-T^{ \sharp_{ A}}\big) \leq w^2_A(T). \end{ eqnarray*} Here $w_A(.), c_A(.)$ and $\|.\|_A $ denote $A$-numerical radius, $A$-Crawford number and $A$-operator seminorm, respectively.
Categories: math.FA