In the following we generalize the concept of Birkhoff–James orthogonality of operators on a Hilbert space when a semi-inner product is considered. More precisely, for linear operators and on a complex Hilbert space , a new relation is defined if and are bounded with respect to the seminorm induced by a positive operator satisfying for all . We extend a theorem due to Bhatia and Šemrl by proving that if and only if there exists a sequence of -unit vectors in such that and . In addition, we give some -distance formulas. Particularly, we prove
Some other related results are also discussed.
"Birkhoff–James orthogonality of operators in semi-Hilbertian spaces and its applications." Ann. Funct. Anal. 10 (3) 433 - 445, August 2019. https://doi.org/10.1215/20088752-2019-0001