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

Abstract Let ???? be a complex Hilbert space and A non-zero positive bounded linear operator on . The main aim of this paper is to discuss general method develop -operator seminorm -numerical radius inequalities semi-Hilbertian operators using the existing corresponding Among many other we prove that if S , T X ? ???? ( ), i.e., -adjoint exist, then 2 ? S ? A X T ? + . $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, 1 4 8 ( ? minsize="2.047em">) c maxsize="1.2em" minsize="1.2em">( minsize="1.2em">) w stretchy="false">( stretchy="false">) $$\begin{align*} & \frac{1}{4}\|T^{\sharp_{A}}T+TT^{\sharp_{A}}\|_A\\ \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg)\\ \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg) +\frac{1}{8}c_A^2\big(T+T^{\sharp_{A}}\big)+\frac{1}{8}c_A^2\big(T-T^{\sharp_{A}}\big)\leq w^2_A(T). \end{align*}$$ Here w (?), c (?) ??? denote radius, -Crawford number seminorm, respectively.