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Smooth rank one perturbations of selfadjoint operators

Authors: S. Hassi, H. S. V. de Snoo and A. D. I. Willemsma
Journal: Proc. Amer. Math. Soc. 126 (1998), 2663-2675
MSC (1991): Primary 47A55, 47A57, 47B25; Secondary 81Q15
MathSciNet review: 1451805
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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.

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Additional Information

S. Hassi
Affiliation: Department of Statistics University of Helsinki PL 54, 00014 Helsinki Finland

H. S. V. de Snoo
Affiliation: Department of Mathematics University of Groningen Postbus 800, 9700 AV Groningen Nederland

Keywords: Rank one perturbation, moments, selfadjoint extension, $Q$-function, Friedrichs extension
Received by editor(s): December 26, 1996
Received by editor(s) in revised form: January 28, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society