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Logarithmic approach to bounded imaginary powers. (English) Zbl 0949.47013
Semi-groupes d’opérateurs et calcul fonctionnel. Ecole d’été, Besançon, France, Juin 1998. Besançon: Université de Franche-Compté et CNRS, Equipe de Mathématiques, Publ. Math. UFR Sci. Tech. Besançon. 16, 121-130 (1998).

The fundamental paper of G. Dore and A. Venni [Math. Z. 196, 189-201 (1987; Zbl 0615.47002)] brought to a rapid development of the theory of operators with bounded imaginary powers. When ${A}^{it}\left(t\in ℝ\right)$ is a ${C}_{0}$-group, its generator is the operator logarithm $ilogA$. It is natural to study operator logarithms and look for a convenient “generation” theorem. The first systematic work on $logA$ was done by V. Nollau [Acta Sci. Math. 30, 161-174 (1969; Zbl 0201.45101)]. The reviewer studied logarithms of Hilbert space operators in [Collect. Math. 45, No. 3, 287-300 (1994; Zbl 0819.47018)] and obtained a characterization of bounded imaginary powers in terms of $logA$. The present article is a resume of the author’s paper “Logarithms and imaginary powers of closed linear operators” [Integral Equations Oper. Theory (to appear)] which represents a helpful contribution to the theory of operator logarithms and the generation of ${A}^{it}$. The author has one more paper on this subject in print: “Logarithmic characterization of bounded imaginary powers” [to appear in the Proceedings of the First International Conference on Semigroups of Operators: Theory and Application, Progress in Nonlinear Differential Equations and Their Applications Vol. 42, Birkhäuser, see also Publ. Math. UFR Sci. Tech. Besançon 16, 121-130 (1998)].

Unfortunately, the author was unaware of the mentioned above paper on the reviewer and repeats some of his results.

##### MSC:
 47A60 Functional calculus of operators 47D06 One-parameter semigroups and linear evolution equations