Spectral theory of compact operators

From Encyclopedia of Mathematics
Revision as of 17:25, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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 .


[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. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article