# Intertwining operator

A continuous linear operator $T: E _ {1} \rightarrow E _ {2}$ such that $T \pi _ {1} ( x) = \pi _ {2} ( x) T$, where $\pi _ {1}$ and $\pi _ {2}$ are mappings of a set $X$ into two topological vector spaces $E _ {1}$ and $E _ {2}$ and $x \in X$. This concept is especially fruitful in the case when $X$ is a group or an algebra and $\pi _ {1} , \pi _ {2}$ are representations of $X$. The set of intertwining operators forms the space $\mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} )$, which is a subspace of the space of all continuous linear mappings from $E _ {1}$ to $E _ {2}$. If $\mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) = ( 0)$ and $\mathop{\rm Hom} ( \pi _ {2} , \pi _ {1} ) = ( 0)$, then $\pi _ {1}$ and $\pi _ {2}$ are called disjoint representations. If $\mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} )$ contains an operator that defines an isomorphism of $E _ {1}$ and $E _ {2}$, then $\pi _ {1}$ and $\pi _ {2}$ are equivalent. If $E _ {1} , E _ {2}$ are locally convex spaces, if $E _ {1} ^ {*}$ and $E _ {2} ^ {*}$ are their adjoints, and if $\pi _ {1} ^ {*}$ and $\pi _ {2} ^ {*}$ are the representations contragredient to $\pi _ {1}$ and $\pi _ {2}$, respectively (cf. Contragredient representation), then for any $T \in \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} )$, the operator $T ^ {*}$ is contained in $\mathop{\rm Hom} ( \pi _ {2} ^ {*} , \pi _ {1} ^ {*} )$. If $\pi _ {1}$ and $\pi _ {2}$ are finite-dimensional or unitary representations and $\pi _ {1}$ is irreducible, then $\pi _ {2}$ admits a subrepresentation equivalent to $\pi _ {1}$ if and only if $\mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) \neq ( 0)$. See also Intertwining number.