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Spectral theory of compact operators

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Riesz theory of compact operators

Let be a complex Banach space and a compact operator on . Then , the spectrum of , is countable and has no cluster points except, possibly, . Every is an eigenvalue, and a pole of the resolvent function . Let be the order of the pole . For each , is closed, and this range is constant for . The null space is finite dimensional and constant for . The spectral projection (the Riesz projector, see Riesz decomposition theorem) has non-zero finite-dimensional range, equal to , and its null space is . Finally, .

The respective integers and are called the index and the algebraic multiplicity of the eigenvalue .

References

[a1] H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) pp. 45ff.
[a2] N. Dunford, J.T. Schwartz, "Linear operators I: General theory" , Interscience (1964) pp. Sect. VII.4
How to Cite This Entry:
Spectral theory of compact operators. M. Hazewinkel (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Spectral_theory_of_compact_operators&oldid=18215
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098