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    STUDIA MATHEMATICA - Issue no. 1 / 2005  
  Article:   R E C E N Z I I.

Authors:  .
  Abstract:  The spectral theory of bounded linear operators on Banach spaces is one of the most important branches of functional analysis and operator theory, with deep and far reaching applications to spectral theory of di erential operators and to classical quantum mechanics. It is expected that a reasonable definition of the spectrum of a continuous nonlinear operator F acting on a Banach space X should agree with the usual one when F is linear and, at a same time, to retain some of its essential properties, as nonemptiness, compactness, to contain the eigenvalues, etc. By a sequence of 8 simple examples given in the introduction the authors show some of the drawbacks of various natural definitions of the spectrum of a nonlinear operator, leading them to the conclusion that the main matter is not the intrinsic structure of the spectrum, but rather its usefulness in the study of nonlinear operator equations. The book contains a systematic presentation of various spectra for nonlinear operators, along with some applications. Numerous examples and tables illustrate the relations between these spectra, as well as some shortcomings arising for each of them.  
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