Hilbert direct sum of unitary representations

Definition

Let $G$ be a topological group, and for each $i\in I$ let $\pi_i:G\to\mathcal U(\mathcal H_i)$ be a strongly continuous unitary representation on a complex Hilbert space. Their Hilbert direct sum is the representation

\bigoplus_{i\in I}\pi_i:G\longrightarrow \mathcal U\left(\bigoplus_{i\in I}\mathcal H_i\right)

defined by

\left(\bigoplus_i\pi_i\right)(g)(\xi_i)_i =\bigl(\pi_i(g)\xi_i\bigr)_i,

where $\bigoplus_i\mathcal H_i$ consists of families satisfying $\sum_i\lVert\xi_i\rVert^2<\infty$ . The coordinatewise operator is unitary and the resulting representation is strongly continuous.

Why continuity is preserved

Every vector in the Hilbert sum can be approximated in norm by vectors with finite support. On a finite-support vector, strong continuity follows from that of finitely many summands. Unitarity gives the uniform bound

\left\lVert\left(\bigoplus_i\pi_i\right)(g)\right\rVert=1,

so the finite-support approximation extends continuity to every vector. This argument works for an arbitrary index set; each individual vector has at most countably many nonzero coordinates.

Examples and decomposition

The sum $\pi\oplus\sigma$ is the basic two-summand example. Repeating one representation $n$ times records finite multiplicity, while a countable sum can record infinite multiplicity. Each coordinate space $\mathcal H_i$ is a closed invariant subspace, and the coordinate projection is an intertwining operator. By contrast, an algebraic direct sum is generally incomplete and is not itself a Hilbert-space representation.

Conventions and scope

The Hilbert direct sum is a discrete construction. It should not be confused with a direct integral of representations, where summands vary over a measure space and vectors are square-integrable measurable fields. No countability assumption on $I$ is required for the definition, although separability of the sum imposes restrictions on the nonzero summands.

References

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