Summary
Vere-Jones [5] gave an example, where the geometric ergodicity of a matrixP has a connection with the spectalproperties of an operatorT p. (For the definition ofT p see below [§ 1]). Here it will be proved that one can find for a geometric ergodic stochstic matrixP an operatorT p with the perperty that in the transient case the spectralradius ofT p is smaller 1 and in the recurrent case the spectrum ofT p lies, except a single point at 1, in a circle with radius smaller 1.
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Literatur
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D. Vere-Jones, On the Spectra of some Linear Operators associated with Queuing Systems, Zschr. Wahrscheinlichkeitstheorie und verw. Gebiete 1963.
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Herrn Prof.Dr. Lothar Collatz zum 60. Geburtstag gewidmet
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Michaliček, J. Ein Zusammenhang zwischen geometrischer Ergodizität und Spektraleigenschaften gewisser Operatoren bei stochastischen Matrizen. Abh.Math.Semin.Univ.Hambg. 36, 166–172 (1971). https://doi.org/10.1007/BF02995919
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DOI: https://doi.org/10.1007/BF02995919