Smooth rank one perturbations of selfadjoint operators

Abstract:

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.