Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators

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.