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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 {\it G. Dore} and {\it 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 $A^{it} (t\in\bbfR)$ is a $C_0$-group, its generator is the operator logarithm $i\log A$. It is natural to study operator logarithms and look for a convenient “generation” theorem. The first systematic work on $\log A$ was done by {\it 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 $\log A$. 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. For the entire collection see [Zbl 0928.00037].

47A60Functional calculus of operators
47D06One-parameter semigroups and linear evolution equations