Schur's lemma for unitary representations - Imported Knowlpedia Content

Statement

Let $\pi$ be a unitary representation of a group $G$ on a nonzero complex Hilbert space $\mathcal H$ . Schur's lemma for unitary representations states that $\pi$ is irreducible if and only if every bounded operator $T$ satisfying

T\pi(g)=\pi(g)T\qquad(g\in G)

is a scalar multiple of the identity. Equivalently, the commutant $\pi(G)'$ equals $\mathbb C I$ . Here irreducibility means that $\mathcal H$ has no nonzero proper closed invariant subspace. No finite-dimensionality, compactness, or local compactness hypothesis on $G$ is required.

Proof mechanism

If a closed subspace is invariant under a unitary representation, its orthogonal complement is invariant, so the corresponding orthogonal projection lies in the commutant. Conversely, for a self-adjoint operator in the commutant, every spectral projection also commutes with $\pi(G)$ . Irreducibility forces all these projections to be $0$ or $I$ , hence the operator is scalar. Applying this to the real and imaginary parts of an arbitrary commutant element proves the theorem Folland, Theorem 3.5.

Intertwiners between irreducibles

Let $\pi$ and $\sigma$ be irreducible unitary representations and let $T$ be a bounded intertwining operator from $\pi$ to $\sigma$ . Then $T=0$ , or $T$ is a positive scalar multiple of a unitary intertwiner. Thus inequivalent irreducibles admit no nonzero bounded intertwiner; if $\pi=\sigma$ , every such $T$ is scalar. This follows by applying the commutant statement to $T^*T$ and $TT^*$ .

Conventions and scope

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §3.1, especially Theorem 3.5.
Jacques Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: Chapter 2 on representations and commutants.