A-numerical radius inequalities for operator matrices in semi-Hilbertian spaces

This paper improves a well-known inequality in the literature by introducing new A-numerical radius inequalities for $2\times 2$ operator matrices. The off-diagonal components of $2\times 2$ operator matrices are one of the primary areas of study, highlighting the special characteristics of smaller matrices in contrast to larger ones. These results improve our understanding of the construction and behavior of operator matrices and extend the theoretical context of A-numerical radius inequalities. In order to illustrate the wider application and potential of the suggested inequalities across different sizes, the $2\times 2$ operator matrices are highlighted in this work. This provides valuable information for further study and real-world applications and represents a major breakthrough in operator theory. Additionally, we provide a few instances that demonstrate the reliability of the results.