# Spectral theory of compact operators

From Encyclopedia of Mathematics

*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