Further A-numerical radius inequalities for semi-Hilbertian space operators

This work aims to introduce a new Buzano-type inequality that integrates and unifies several well-established results from the literature. As a consequence, we present novel numerical radius bounds for operators in semi-Hilbertian spaces. For example, it is proven that for T is an element of LA(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T \in \mathcal {L}_{A}(\mathcal {H})$$\end{document} and a mapping chi:[0,1]subset of J ->[14,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi : [0,1]\subset J \rightarrow [\frac{1}{4},1]$$\end{document}, omega A4(T)<=chi(lambda)4T & sharp;AT+TT & sharp;AA2+(1-chi(lambda))2T & sharp;AT+TT & sharp;AA omega AT2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega _{A}<^>{4}(T) \leqslant \frac{\chi (\lambda )}{4}\left\| T<^>{\sharp _{A}} T+TT<^>{\sharp _{A}}\right\| _{A}<^>{2}+\frac{(1-\chi (\lambda ))}{2}\left\| T<^>{\sharp _{A}} T+T T<^>{\sharp _{A}}\right\| _{A} \omega _{A}\left( T<^>{2}\right) . \end{aligned}$$\end{document}Additionally, we establish several bounds for the A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {A}$$\end{document}-numerical radii of 2x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \times 2$$\end{document} operator matrices. Our results extend and improve certain well-established inequalities from existing literature.