Intertwining operator between unitary representations

Definition

Let $(\pi,\mathcal H_\pi)$ and $(\sigma,\mathcal H_\sigma)$ be strongly continuous unitary representations of the same group $G$ . An intertwining operator from $\pi$ to $\sigma$ is a bounded linear operator $T:\mathcal H_\pi\to\mathcal H_\sigma$ satisfying

T\pi(g)=\sigma(g)T\qquad\text{for every }g\in G.

The vector space of all such operators is denoted $\operatorname{Hom}_G(\pi,\sigma)$ . Boundedness is part of the Hilbert-representation definition; an everywhere-defined algebraic operator satisfying the displayed identity need not be continuous.

Basic properties

Intertwiners compose, identity operators intertwine a representation with itself, and the adjoint $T^*$ intertwines $\sigma$ with $\pi$ . The kernel of $T$ and the closure of its range are invariant closed subspaces. Consequently, intertwiners are the morphisms in the category of unitary representations with bounded equivariant maps.

Equivalence and irreducibility

A unitary intertwiner that is onto exhibits unitary equivalence of the two representations. For irreducible complex unitary representations, a nonzero intertwiner is a scalar multiple of a unitary equivalence; in particular, the commutant $\operatorname{Hom}_G(\pi,\pi)$ consists of scalars. This is the Hilbert-space form of Schur's lemma, using polar decomposition and closed invariant subspaces Folland, §3.1.

Conventions and scope

In smooth or distribution representation theory, “intertwining operator” may mean a continuous map for a locally convex topology, or a densely defined unbounded operator with an invariant domain. Those notions require the topology or domain to be stated and are not covered by this bounded Hilbert-space definition.

References

G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §3.1 on intertwiners, equivalence, and irreducibility.