An Extension of Birkhoff–James Orthogonality Relations in Semi-Hilbertian Space Operators

Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given positive operator $A\in\B(\h)$, and number $\lambda\in [0,1]$, seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined set $\B_{A^{1/2}}(\h)$ in $\B(\h)$ having an $A^{1/2}$-adjoint. The combination sesquilinear form ${\langle \cdot, \cdot\rangle}_A$ its induced ${\|\cdot\|}_A$. A characterization Birkhoff--James orthogonality for with respect to discussed given. Moving $\lambda$ along interval $[0,1]$, wide spectrum seminorms are obtained, $A$-numerical radius $w_A(\cdot)$ at beginning (associated $\lambda=0$) $A$-operator ${\|\cdot\|}_A$ end $\lambda=1$). Moreover, if $A=I$ identity operator, classical norm numerical obtained. Therefore, results this paper significant extensions generalizations known area.