On rank-one perturbations in semi-Hilbertian spaces

Let A be a positive bounded linear operator on a Hilbert space (H, ⟨·, ·⟩) and ⟨u, v⟩A:= ⟨Au, v⟩, u, v ∈ H the induced semi-inner product. In this paper, we study rank one perturbations of bounded linear operators in the semi-Hilbertian space (H, ⟨·, ·⟩A), R = T + u ⊗ (Av), for u, v ∈ H (A). In addition to proving some properties of these operators related to (A, m)-isometries and (A, n)-symmetries, we characterize some conditions for R to be A-hyponormal or A-normal.