Smooth rank one perturbations of selfadjoint operators

Let $A$ be a selfadjoint operator in a Hilbert space ${\mathfrak H}$ with inner product $[\cdot,\cdot]$. The rank one perturbations of $A$ have the form $A+\tau[\cdot,\omega] \omega$, $\tau \in {\mathbb R}$, for some element $\omega \in {\mathfrak H}$. In this paper we consider smooth perturbations, i.e. we consider $\omega \in \operatorname{dom}|A|^{k/2}$ for some $k \in {\mathbb N}\cup \{0\}$. Function-theoretic properties of their so-called $Q$-functions and operator-theoretic consequences will be studied.