Birkhoff–James orthogonality of operators in semi-Hilbertian spaces and its applications

Abstract

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 T and S on a complex Hilbert space H, a new relation T⊥ABS is defined if T and S are bounded with respect to the seminorm induced by a positive operator A satisfying ‖T+γS‖A≥‖T‖A for all γ∈C. We extend a theorem due to Bhatia and Šemrl by proving that T⊥ABS if and only if there exists a sequence of A-unit vectors {xn} in H such that lim n→+∞‖Txn‖A=‖T‖A and lim n→+∞〈Txn,Sxn〉A=0. In addition, we give some A-distance formulas. Particularly, we prove inf γ∈C‖T+γS‖A=sup {|〈Tx,y〉A|;‖x‖A=‖y‖A=1,〈Sx,y〉A=0}. Some other related results are also discussed.