Krylov-Bogolyubov substitution in the perturbation theory of linear operators (Q1081818)

This paper intends to clarify the role of the Krylov-Bogolyubov transformation [\textit{Yu. A. Mitropol'skij}, The Method of Averaging in Nonlinear Mechanics (in Russian) (1971; Zbl 0325.70002)] in the study of the spectral properties of perturbed operators \(A-\epsilon B,0<\epsilon <\delta\), of a closed linear operator A, where the B are A-bounded operators. In particular, it is shown that by operator-theoretically reformulating some methods used in celestial mechanics and nonlinear vibration theory, it is reduced to the method of similar operators. Namely, under some abstract conditions on A and B, the operators A- \(\epsilon\) B can be transformed to more tractable ones \(A-\epsilon JB- \epsilon B_ 1(\epsilon)\), with J a projection and \(\lim_{\epsilon \to 0}B_ 1(\epsilon)=0\), by similarity transformations U(\(\epsilon)\), \(0<\epsilon \leq \delta\), for a \(\delta\) sufficiently small.