Journal of the Mathematical Society of Japan

Spectrum for compact operators on Banach spaces


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For a two-sided sequence of compact linear operators acting on a Banach space, we consider the notion of spectrum defined in terms of the existence of exponential dichotomies under homotheties of the dynamics. This can be seen as a natural generalization of the spectrum of a matrix—the set of its eigenvalues. We give a characterization of all possible spectra and explicit examples of sequences for which the spectrum takes a form not occurring in finite-dimensional spaces. We also consider the case of a one-sided sequence of compact linear operators.


The first author and the third author were supported by FCT/Portugal through UID/MAT/04459/2013. The second author was supported in part by the Croatian Science Foundation under the project IP-2014-09-2285 and by the University of Rijeka under the project number

Article information

J. Math. Soc. Japan, Volume 71, Number 1 (2019), 1-17.

Received: 25 October 2016
Revised: 12 May 2017
First available in Project Euclid: 14 September 2018

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Primary: 37D99: None of the above, but in this section

compact operators exponential dichotomies spectra


BARREIRA, Luis; DRAGIČEVIĆ, Davor; VALLS, Claudia. Spectrum for compact operators on Banach spaces. J. Math. Soc. Japan 71 (2019), no. 1, 1--17. doi:10.2969/jmsj/76447644.

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