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

Abstract

Let ${\mathbb {B}}({\mathcal {H}})$ denote the $C^{*}$-algebra of all bounded linear operators on a Hilbert space $\big ({\mathcal {H}}, \langle \cdot , \cdot \rangle \big )$. Given a positive operator $A\in {\mathbb {B}}({\mathcal {H}})$, and a number $\lambda \in [0,1]$, a seminorm ${\Vert \cdot \Vert }_{(A,\lambda )}$ is defined on the set ${\mathbb {B}}_{A^{1/2}}({\mathcal {H}})$ of all operators in ${\mathbb {B}}({\mathcal {H}})$ having an $A^{1/2}$-adjoint. The seminorm ${\Vert \cdot \Vert }_{(A,\lambda )}$ is a combination of the sesquilinear form ${\langle \cdot , \cdot \rangle }_A$ 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.