On Rank-One Perturbations in Semi-Hilbertian Spaces

Abstract

Let $A$ be a positive bounded linear operator on a Hilbert space $(\mathcal{H},\langle\cdot\;,\,\cdot\rangle)$ and $\langle u, v\rangle_A := \langle Au, v\rangle,\; u,\,v \in \mathcal{H}$ the induced semi-inner product. In this paper, we study rank one perturbations of bounded linear operators in the semi-Hilbertian space $(\mathcal{H},\langle\cdot\;,\,\cdot\rangle_A)$, $R=T+u\otimes (Av)$, for $u,\,v\in \mathcal{H}\backslash \mathcal{N}(A)$. 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.