Aleksandrov, A. B.; Peller, V. V. Functions of perturbed dissipative operators. (English. Russian original) Zbl 1250.47013 St. Petersbg. Math. J. 23, No. 2, 209-238 (2012); translation from Algebra Anal. 23, No. 2, 9-51 (2011). In this paper, the authors continue their previous studies of functions of perturbed operators, extending the results that had been obtained for self-adjoint, unitary, or contractive operators, to dissipative operators. The proof techniques are based on double operator integrals and multiple operator integrals with respect to semispectral measures, but pose a higher level of difficulty than the previously treated cases. The paper is organized into 11 sections and presents a very interesting study of the topic. After an introduction, in the second section the Besov classes and spaces \(A_\omega(\mathbb{R})\) and \(A_{\omega,m}(\mathbb{R})\) are presented, which play an important role in solving perturbation theory problems. Also, a brief introduction to double and multiple operator integrals and dissipative operators is given in the third and fourth sections. It is known that a Lipschitz function is not necessarily operator Lipschitz, i.e., \(|f(x)-f(y)|\leq \text{const}|x-y|\) does not imply \(\|f(A)-f(B)\|\leq \text{const}\,\|A-B\|\) even for self-adjoint operators \(A,B\) on a Hilbert space. Previously, necessary conditions were found, but in the fifth section of the paper sharp conditions for functions analytic in the upper half-plane to be operator Lipschitz and operator differentiable are obtained. Hilbert-Schmidt Lipschitz functions are characterized in the sixth section, and the seventh section is devoted to Hölder functions and functions of class \(A_\omega(\mathbb{R})\). Also, in the eighth section estimates for higher order operator differences in terms of multiple operator integrals for Hölder-Zygmund classes and for the spaces \(A_{\omega,m}(\mathbb{R})\) are obtained. Conditions under which higher operator derivatives exist in terms of multiple operator integrals are studied in the ninth section. In Section 10, similar results as obtained for self-adjoint operators in [A. B. Aleksandrov and V. V. Peller, “Functions of operators under perturbations of class \(S_p\)”, J. Funct. Anal. 258, No. 11, 3675–3724 (2010; Zbl 1196.47012)] are generalized. In the last section of the paper, estimates for commutators \(f(L)R-Rf(M)\) in terms of \(LR-RM\), where \(L\) and \(M\) are maximal dissipative operators and \(R\) is a bounded operator, are obtained. Reviewer: Ilie Valuşescu (Bucureşti) Cited in 14 Documents MSC: 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) 47B44 Linear accretive operators, dissipative operators, etc. Keywords:dissipative operators; perturbations of operators; Schatten-von Neumann classes; Hölder-Zygmund spaces; Besov spaces; continuity moduli Citations:Zbl 1196.47012 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Aleksei Aleksandrov and Vladimir Peller, Functions of perturbed operators, C. R. Math. Acad. Sci. Paris 347 (2009), no. 9-10, 483 – 488 (English, with English and French summaries). · Zbl 1168.47011 · doi:10.1016/j.crma.2009.03.004 [2] A. B. Aleksandrov and V. V. Peller, Operator Hölder-Zygmund functions, Adv. Math. 224 (2010), no. 3, 910 – 966. · Zbl 1193.47017 · doi:10.1016/j.aim.2009.12.018 [3] A. B. Aleksandrov and V. V. 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