## Smooth rank one perturbations of selfadjoint operators

HTML articles powered by AMS MathViewer

- by S. Hassi, H. S. V. de Snoo and A. D. I. Willemsma PDF
- Proc. Amer. Math. Soc.
**126**(1998), 2663-2675 Request permission

## 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.## References

- N. I. Akhiezer,
*The classical moment problem and some related questions in analysis*, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR**0184042** - S. Albeverio and P. Kurasov, "Rank one perturbations of not semibounded operators", Integral Equations Operator Theory 27 (1997), 379â€“400.
- Earl A. Coddington and Hendrik S. V. de Snoo,
*Positive selfadjoint extensions of positive symmetric subspaces*, Math. Z.**159**(1978), no.Â 3, 203â€“214. MR**500265**, DOI 10.1007/BF01214571 - William F. Donoghue Jr.,
*Monotone matrix functions and analytic continuation*, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR**0486556**, DOI 10.1007/978-3-642-65755-9 - F. Gesztesy and B. Simon,
*Rank-one perturbations at infinite coupling*, J. Funct. Anal.**128**(1995), no.Â 1, 245â€“252. MR**1317717**, DOI 10.1006/jfan.1995.1030 - S. Hassi, M. KaltenbĂ¤ck, and H.S.V. de Snoo, "Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass $\textbf {N}_1$ of Nevanlinna functions", J. Operator Theory, 37 (1997), 155-181.
- S. Hassi, M. KaltenbĂ¤ck, and H.S.V. de Snoo, "A characterization of semibounded selfadjoint operators", Proc. Amer. Math. Soc. 125 (1997), 2681â€“2692.
- S. Hassi, H. Langer, and H.S.V. de Snoo, "Selfadjoint extensions for a class of symmetric operators with defect numbers $(1,1)$", Topics in Operator Theory, Operator Algebras and Applications (Timisoara, 1994), Romanian Acad., Bucharest, 1995, pp. 115â€“145.
- S. Hassi and H. S. V. de Snoo,
*On some subclasses of Nevanlinna functions*, Z. Anal. Anwendungen**15**(1996), no.Â 1, 45â€“55. MR**1376588**, DOI 10.4171/ZAA/687 - S. Hassi and H.S.V. de Snoo, "One-dimensional graph perturbations of selfadjoint relations", Ann. Acad. Sci. Fenn., Series A.I. Math., 22 (1997), 123-164.
- S. Hassi and H.S.V. de Snoo, "Nevanlinna functions, perturbation formulas, and triplets of Hilbert spaces", Math. Nachr., (to appear).
- Cahit Arf,
*Untersuchungen ĂĽber reinverzweigte Erweiterungen diskret bewerteter perfekter KĂ¶rper*, J. Reine Angew. Math.**181**(1939), 1â€“44 (German). MR**18**, DOI 10.1515/crll.1940.181.1 - I.S. Kac and M.G. KreÄn, "$R$-functionsâ€“analytic functions mapping the upper halfplane into itself", Supplement I to the Russian edition of F.V. Atkinson,
*Discrete and continuous boundary problems*, Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl., (2) 103 (1974), 1-18). - A. Kiselev and B. Simon,
*Rank one perturbations with infinitesimal coupling*, J. Funct. Anal.**130**(1995), no.Â 2, 345â€“356. MR**1335385**, DOI 10.1006/jfan.1995.1074 - Martin Jurchescu,
*Riemann surfaces and holomorphic mappings*, Acad. R. P. RomĂ®ne. Stud. Cerc. Mat.**12**(1961), 575â€“590 (Romanian, with English and Russian summaries). MR**131542** - Barry Simon,
*Spectral analysis of rank one perturbations and applications*, Mathematical quantum theory. II. SchrĂ¶dinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp.Â 109â€“149. MR**1332038**, DOI 10.1090/crmp/008/04 - A. V. Shtraus,
*Generalized resolvents of nondensely defined bounded symmetric operators*, Functional analysis, No. 27 (Russian), Ulâ€˛yanovsk. Gos. Ped. Inst., Ulâ€˛yanovsk, 1987, pp.Â 187â€“196 (Russian). MR**1129513**

## Additional Information

**S. Hassi**- Affiliation: Department of Statistics University of Helsinki PL 54, 00014 Helsinki Finland
- Email: hassi@cc.helsinki.fi
**H. S. V. de Snoo**- Affiliation: Department of Mathematics University of Groningen Postbus 800, 9700 AV Groningen Nederland
- Email: desnoo@math.rug.nl
- Received by editor(s): December 26, 1996
- Received by editor(s) in revised form: January 28, 1997
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 2663-2675 - MSC (1991): Primary 47A55, 47A57, 47B25; Secondary 81Q15
- DOI: https://doi.org/10.1090/S0002-9939-98-04335-4
- MathSciNet review: 1451805