An extension of Birkhoff--James orthogonality relations in semi-Hilbertian space operators

Abstract

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