Unitary equivalence of representations - Imported Knowlpedia Content

Definition

Let $\pi_1:G\to U(H_1)$ and $\pi_2:G\to U(H_2)$ be strongly continuous unitary representations of the same topological group. They are unitarily equivalent if there is a unitary operator $U:H_1\to H_2$ such that

U\pi_1(g)=\pi_2(g)U\qquad\text{for every }g\in G.

Thus a unitary equivalence is an invertible intertwining operator that also preserves inner products. It identifies the two actions after an isometric change of Hilbert-space coordinates, rather than asserting that their operators are literally equal on one fixed space.

Invariants

A unitary equivalence carries closed invariant subspaces of $H_1$ to closed invariant subspaces of $H_2$ . It therefore preserves irreducibility, orthogonal direct-sum decompositions, multiplicities, and coefficient functions after vectors are transported by $U$ . The integrated representations also have the same operator norms and corresponding kernels.

Equivalence relation and classes

Identity operators, adjoints, and compositions show that unitary equivalence is an equivalence relation. Representation theory normally classifies unitary representations up to this relation. In particular, the unitary dual of a locally compact group consists of unitary-equivalence classes of irreducible unitary representations; see Folland, §3.1.

Conventions and scope

For unitary representations, a bounded invertible intertwiner does imply unitary equivalence: the unitary factor in its polar decomposition still intertwines the actions. The original intertwiner itself need not be unitary. Equality of characters or of selected coefficient functions does not, without an appropriate classification theorem, constitute the definition of unitary equivalence.

References

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