ISSN:
1420-8938
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. Let L be a monic operator polynomial on a Banach space X. For a compact set $ \sigma $ of complex numbers X (L, $ \sigma $ ) denotes the set of all vectors x such that $ L^{-1}(\,\cdot\,)\,x $ has a holomorphic extension to the complement of $ \sigma $ . A relative open-and-closed subset $ \sigma $ of $ \sigma (L) $ (the spectrum of L) is called a spectral set of L if X (L, $ \sigma $ ) + X (L, $ \sigma (L) \setminus \sigma )$ = X. The main characterization reads as follows: A set is a spectral set of L iff the corresponding spectral projection of the companion operator is block diagonal with identical projections as blocks. These projections have a contour integral representation and their ranges have the covenient properties reminiscent to the spectral theory of linear operators. The special cases of a singleton or a pole being a spectral set is characterized; in these cases certain special Jordan chains exist. Further, the connection between the spectral sets of two polynomials and the spectral sets of their product is studied.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s000130050091
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