ISSN 1066-369X, Russian Mathematics, 2017, Vol. 61, No. 12, pp. 19–28. c Allerton Press, Inc., 2017. Original Russian Text c A.R. Mirotin, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 12, pp. 24–34.

A Functional Calculus of Closed Operators on Banach Space. III. Certain Topics of Perturbation Theory

A. R. Mirotin1* 1F. Skorina Gomel State University ul. Sovetskaya 104, Gomel, 246019 Republic of Belarus Received August 10, 2016

Abstract—The present paper is a continuation of research of A. A. Atvinovskii and of the author in the area of functional calculus of closed operators on Banach spaces based on Markov and related functions as symbols. The following topics in the perturbation theory are considered: Estimates of bounded perturbations of operator functions with respect to general operator ideal norms, Lipschitz property, moment inequality, Frechet´ differentiability, analyticity of operator functions under consid- eration with respect to the perturbation parameter, spectral shift function, and Lifshits–Krein trace formula. DOI: 10.3103/S1066369X17120039 Keywords: perturbation of operator, operator Lipschitzness, moment inequality, operator differentiability, spectral shift function, Lifshits–Krein trace formula.

INTRODUCTION AND PRELIMINARIES There exists a great body of publications (for instance, [1–5]) dealing with various problems of theory of perturbations of linear operators, which are related to classical functional von Neumann–Mourier– Danford calculus of self-adjoint, unitary and normal operators in Hilbert spaces. The present paper treats analogous questions of theory of perturbations of linear operators arising in functional calculus of closed operators in Banach spaces built in [6, 7] (see also [8, 9]). But the present paper is independent of the mentioned works. We consider here the following items: estimates for bounded perturbations of operator functions in norms of general operator ideals, operator and commutator Lipschitz properties of these functions, moment inequality, Frechet´ operator differentiability, analytic dependence of meanings of operator functions under consideration on the perturbation parameter, function of spectral shift, and analog of the Lifshits–Krein trace formula for nuclear perturbations. The author knows only one paper on Lipschitz operator and commutator functions in the context of Banach spaces. It is the recent paper [10]1). The subject of this paper is functional calculus for operators of scalar type in Banach spaces using as symbols a rather broad class of functions connected closely with 1,2 R ˙ 1 R the Sobolev space W ( ) and the homogeneous Besov space B∞,1( ). Particularly, in the mentioned paper a theory of double operator integrals is developed for operators of scalar type in Banach spaces, and by means of this theory there are obtained estimates of commutators of rather general kind in the operator norm and in the norms of operator ideals satisfying certain intrinsic restrictions. Let us recall necessary concepts and results from [6] and [7].

Definition 1. Afunctiong belongs to the Markov class R[a, b], a

*E-mail: [email protected]. 1)The author is grateful to the referee who indicated to him this work.

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