Abstract
In the paper one investigates nondissipative operators L in a Hilbert space; for them one constructs a model representation similar to the Nagy-Foias model for dissipative operators. In this representation one succeeds to calculate the action of the resolvent of a nondissipative operator on selected subspaces. This allows us to relinquish the consideration of the
-self-adjoint dilation of the operator, whose spectral representation involves considerable difficulties. Isolated results are new also for the dissipative case which is not excluded. In part I one considers the “triangular” factorization of the characteristic funcion of the operator L and one carries out the proof of the fundamental theorem which gives a formula for the calculation of (L-No)−1(JmNoO TmNo<O) on selected subspaces. The applications of this theorem to the spectral analysis on the absolutely continuous spectrum and to the problems of linear similitude for nondissipative operators are considered in part II.
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Ihstituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 90–102, 1976.
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Naboko, S.N. Absolutely continuous spectrum of a nondissipative operator and the functional model. I. J Math Sci 16, 1109–1117 (1981). https://doi.org/10.1007/BF02427720
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DOI: https://doi.org/10.1007/BF02427720