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Finite-rank spectral pieces of a self-adjoint compact convolution operator

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $K$ be a compact Hausdorff group with normalized Haar probability $μ$ and let $φ \in L^{2} (K, μ; C)$ satisfy $φ^{*} = φ$ , so $C_{φ}$ is compact and self-adjoint (Compact convolution operators commute with right translations and have conjugate-kernel adjoints, L² convolution on a compact group is Hilbert–Schmidt). Let $Σ = {λ \neq 0 : λ is an eigenvalue of C_{φ}}$ and $E_{λ} = \ker (C_{φ} - λ I)$ .

$Σ$ is finite or countably infinite, each $E_{λ}$ is finite dimensional, distinct eigenspaces are orthogonal, and the closed linear span of $⋃_{λ \in Σ} E_{λ}$ is $(\ker C_{φ})^{⊥}$ , so $L^{2} (K) = \ker C_{φ} \oplus {⨁^{^}}_{λ \in Σ} E_{λ}$ is an orthogonal Hilbert-space direct sum.
Every $E_{λ}$ ( $λ \in Σ$ ) and $\ker C_{φ}$ are invariant under the right regular representation $ρ$ ; consequently each $E_{λ}$ is a finite-dimensional continuous unitary representation of $K$ under $ρ$ .

Facts & Assumptions

[F1]

The spectral theorem for a compact self-adjoint operator $T$ on a Hilbert space $H$ : the set $Σ$ of nonzero eigenvalues is finite or countably infinite, each eigenspace $E_{λ} = \ker (T - λ I)$ is finite dimensional, the closed linear span $M$ of $⋃_{λ \in Σ} E_{λ}$ equals $(\ker T)^{⊥}$ , and $H = M \oplus M^{⊥}$ with $M^{⊥} \subseteq \ker T$ . (Spectral theorem for compact self adjoint operators)

[F2]

Eigenspaces of a self-adjoint operator belonging to distinct eigenvalues are orthogonal. (Eigenspaces of a self adjoint operator are orthogonal)

[F3]

Under $φ^{*} = φ$ the convolution operator $C_{φ}$ is compact and self-adjoint, and each eigenspace $\ker (C_{φ} - λ I)$ and the kernel $\ker C_{φ}$ are invariant under the right regular representation $ρ$ . (Compact convolution operators commute with right translations and have conjugate-kernel adjoints, L² convolution on a compact group is Hilbert–Schmidt)

[F4]

For pairwise orthogonal closed subspaces whose closed linear span is $H$ , the canonical map from their Hilbert direct sum onto $H$ is a unitary intertwiner; and a representation that restricts to representations on such a family of $π (K)$ -invariant subspaces is the Hilbert direct sum of those subrepresentations. (Hilbert direct sums of unitary representations)

[F5]

Every closed linear subspace $M$ of a Hilbert space satisfies $H = M \oplus M^{⊥}$ . (Orthogonal decomposition by a closed subspace, Orthogonality and the orthogonal complement)

[F6]

The right regular representation $ρ$ is a strongly continuous unitary representation of $K$ on $L^{2} (K)$ , and its restriction to a closed invariant subspace is again a strongly continuous unitary representation. (The regular representations are unitary, strongly continuous, and the left one is faithful, Left and right regular unitary representations of an LCH group)

Proof

Given: AC, a compact Hausdorff group $K$ with normalized Haar probability $μ$ , a class $φ \in L^{2} (K)$ with $φ^{*} = φ$ , the compact self-adjoint operator $C_{φ}$ , and the set $Σ$ of its nonzero eigenvalues with eigenspaces $E_{λ}$ .

1.1F1F2F4F5

The spectral theorem [F1] applied to $T = C_{φ}$ gives that $Σ$ is finite or countably infinite, that every $E_{λ}$ is finite dimensional, and that the closed linear span $M$ of $⋃_{λ \in Σ} E_{λ}$ equals $(\ker C_{φ})^{⊥}$ ; distinct eigenspaces are orthogonal by [F2]; [F5] applied to the closed subspace $M$ gives $L^{2} (K) = M \oplus M^{⊥}$ with $M = (\ker C_{φ})^{⊥}$ , so $M^{⊥} = \ker C_{φ}$ and hence $L^{2} (K) = \ker C_{φ} \oplus M$ as an orthogonal decomposition; since the $E_{λ}$ are pairwise orthogonal closed subspaces with closed linear span $M$ , the canonical map ${⨁^{^}}_{λ \in Σ} E_{λ} \to M$ is a unitary isomorphism by [F4], so $L^{2} (K) = \ker C_{φ} \oplus {⨁^{^}}_{λ \in Σ} E_{λ}$ is an orthogonal Hilbert-space direct sum; this is (1).

2.1F3F4F6step 1.1∎

By the choice $φ^{*} = φ$ and [F3], every $E_{λ}$ and $\ker C_{φ}$ is invariant under the right regular representation $ρ$ ; each $E_{λ}$ is closed and finite dimensional by step 1.1, and by [F6] the restriction of $ρ$ to $E_{λ}$ is a strongly continuous unitary representation of $K$ , hence a finite-dimensional continuous unitary representation; this is (2), and it also exhibits $ρ$ as the Hilbert direct sum of these subrepresentations and the kernel by [F4]. The Axiom of Choice is consumed through the normalized Haar probability, the compact self-adjoint spectral theorem and the cited Hilbert-space suppliers; the argument above is choice-free apart from those inputs.

Finite-rank spectral pieces of a self-adjoint compact convolution operator

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources