# Intertwining operator

A continuous linear operator such that , where and are mappings of a set into two topological vector spaces and and . This concept is especially fruitful in the case when is a group or an algebra and are representations of . The set of intertwining operators forms the space , which is a subspace of the space of all continuous linear mappings from to . If and , then and are called disjoint representations. If contains an operator that defines an isomorphism of and , then and are equivalent. If are locally convex spaces, if and are their adjoints, and if and are the representations contragredient to and , respectively (cf. Contragredient representation), then for any , the operator is contained in . If and are finite-dimensional or unitary representations and is irreducible, then admits a subrepresentation equivalent to if and only if . See also Intertwining number.

#### References

[1] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) |

[2] | A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian) |

**How to Cite This Entry:**

Intertwining operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Intertwining_operator&oldid=47402