$A$-Numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications

In this paper, we aim to introduce and characterize the concept of numerical radius orthogonality of operators on a complex Hilbert space $\mathcal{H}$ which are bounded with respect to the semi-norm induced by a positive operator $A$ on $\mathcal{H}$. Moreover, a characterization of the $A$-numerical radius parallelism for $A$-rank one operators is proved. As applications of the obtained results, we obtain some $\mathbb{A}$-numerical radius inequalities of operator matrices where $\mathbb{A}$ is the operator diagonal matrix with diagonal entries are positive operator $A$. Some other related results are also investigated.