Selected problems in perturbation theory (Q2906464)

In this survey, based on a series of articles, the author discusss several selected problems in perturbation theory, mostly arising in connection with the study of the behaviour of functions of operators. Specifically, these refer to the behaviour of functions under smooth perturbations.NEWLINENEWLINEThe article begins with a brief introduction in the theory of double operator integrals developed by \textit{M. Sh. Birman} and \textit{M. Z. Solomyak} [Probl. Mat. Fiz. 1, 33--67 (1966; Zbl 0161.34602); ibid. 2, 26--60 (1967; Zbl 0182.46202); ibid. 6, 27--53 (1973; Zbl 0281.47013); Integral Equations Oper. Theory 47, No. 2, 131--168 (2003; Zbl 1054.47030)]. It then goes on with introductory material concerning Besov spaces, starting with Besov spaces on the unit circle, for which different descriptions are given. The Hölder-Zygmund scale is also described. For the case of the real line, the discussion focusses on the description of homogeneous Besov spaces. For more detailed information on Besov spaces and their applications, the reader is referred to [\textit{J. Peetre}, New thoughts on Besov spaces. Durham: Duke University (1976; Zbl 0356.46038); \textit{V. V. Peller}, Hankel operators and their applications. New York: Springer (2003; Zbl 1030.47002)].NEWLINENEWLINEAfter some remarks concerning the nuclearity properties of Hankel operators borrowed from \textit{V. V. Peller} [Funct. Anal. Appl. 19, 111-123 (1985); translation from Funkts. Anal. Prilozh. 19, No. 2, 37--51 (1985; Zbl 0587.47016)] and the author's own aforementioned book, the topic shifts further to Lipschitz operators and to the problem of differentiability of functions of self-adjoint operators on a Hilbert space. In particular, the behaviour of functions under perturbations of an operator by operators of Schatten-von Neumann classes is discussed. The applicability of the presented results to the Livshits-Krein and Koplienko-Neidhardt trace formulae is also shown.NEWLINENEWLINEThe next topic concerns operator Hölder-Zygmund functions. The author recalls results originally obtained in [\textit{A. B. Aleksandrov} and \textit{V. V. Peller}, C. R., Math., Acad. Sci. Paris 347, No. 9--10, 483--488 (2009; Zbl 1168.47011); Adv. Math. 224, No. 3, 910--966 (2010; Zbl 1193.47017)] and presents improvements of related results by \textit{Yu. B. Farforovskaya} [Vestn. Leningr. Univ. 23, No.19, Ser. Mat. Mekh. Astron. 23, No. 4, 94--97 (1968; Zbl 0165.48002)]. The last results touch on extensions of considered problems to perturbations of normal operators.NEWLINENEWLINEThe reader will find this article a useful introduction to the mentioned problems and a good starting point for further studies.NEWLINENEWLINEFor the entire collection see [Zbl 1232.30005].