98-413 V. Baladi and M. Holschneider

Approximation of the nonessential spectrum of transfer operators (52K, AMS TeX) Jun 5, 98

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Abstract. We give sufficient conditions to approximate the ``nonessential'' spectrum of a bounded operator L acting on a Banach space B by part of the spectra of a sequence of compact (or finite rank) operators L_j=(I-P_j) L (I-P_j), where $I-P_j is a suitable family of uniformly bounded operators which approach the identity. (By nonessential spectrum we mean here all the spectrum outside of the disc of radius equal to the essential spectral radius.) For this, we combine formulas for the essential spectral radius in terms of the P_j with nonstandard perturbative results on the stability of the nonessential spectrum of quasicompact operators. We present concrete applications to transfer operators of smooth expanding maps using multiresolution analysis (large scale approximation projections).

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