# Compact operator

An operator defined on a subset of a topological vector space with values in a topological vector space , such that every bounded subset of is mapped by it into a pre-compact set (cf. Pre-compact space) of . If, in addition, the operator is continuous on , then it is called completely continuous on this set. In the case when and are Banach or, more generally, bornological spaces and the operator is linear, the concept of a compact and a completely-continuous operator are the same. If is a compact and is a continuous operator, then and are compact operators, so that the set of compact operators is a two-sided ideal in the ring of all continuous operators. In particular, a compact operator does not have a continuous inverse. The property of compactness plays an essential role in the theory of fixed points of an operator and in the study of its spectrum, which in this case has a number of "good" properties.

Examples of compact operators are the Fredholm integral operators (cf. Integral operator)

the Hammerstein operator

and the Urysohn (Uryson) operator

in certain function spaces, under suitable restrictions on the functions , and .

#### References

[1] | L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian) |

[2] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 |

[3] | W. Rudin, "Functional analysis" , McGraw-Hill (1973) |

[4] | M.A. Krasnosel'skii, et al., "Integral operators and spaces of summable functions" , Noordhoff (1976) (Translated from Russian) |

#### Comments

#### References

[a1] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |

[a2] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |

**How to Cite This Entry:**

Compact operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Compact_operator&oldid=13410