Certain Upper Bounds on the $A$-Numerical Radius of Operators in Semi-Hilbertian Spaces and Their Applications

Consider a complex Hilbert space $\left(\mathcal{H}, \langle \cdot, \cdot \rangle\right)$ equipped with a positive bounded linear operator $A$ on $\mathcal{H}$. This induces a semi-norm $\|\cdot\|_A$ through the semi-inner product $\langle x, y \rangle_A = \langle Ax, y \rangle$ for $x, y \in \mathcal{H}$. In this semi-Hilbertian space setting, we investigate the $A$-numerical radius $w_A(T)$ and $A$-operator semi-norm $\|T\|_A$ of bounded operators $T$. This paper presents significant improvements to existing upper bounds for the $A$-numerical radius of operators in semi-Hilbertian spaces. We establish several new inequalities that provide sharper estimates than those currently available in the literature. Notably, we refine the triangle inequality for the $A$-operator semi-norm, offering more precise characterizations of operator behavior. To validate our theoretical advancements, we provide concrete examples that demonstrate the effectiveness of our results. These examples illustrate how our bounds offer tighter estimates compared to previous ones, confirming the practical relevance of our findings. Our work contributes to the broader understanding of numerical radius inequalities and their applications in operator theory, with potential implications for functional analysis and related fields.