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Bounded multipliers on compact self-adjoint eigenspaces define bounded operators

Statement

Assume Countable Choice. Let $F \in {R, C}$ , $H$ be an $F$ -Hilbert space, and let $S : H \to H$ be compact and self-adjoint. Let $Σ$ be its set of nonzero eigenvalues, $E_{t} = \ker (S - t I)$ , $P_{t}$ the orthogonal projection onto $E_{t}$ , and $P_{0}$ the orthogonal projection onto $\ker S$ . If $q : {0} \cup Σ \to F$ is bounded, then for every $x \in H$ the finite-subset net $q (0) P_{0} x + \sum_{t \in F} q (t) P_{t} x, F \subseteq Σ finite,$ converges in norm to a vector $Q_{q} x$ . The map $Q_{q} : H \to H$ is bounded and linear, with $∥ Q_{q} ∥ \leq \sup_{t \in {0} \cup Σ} ∣ q (t) ∣$ .

Facts & Assumptions

Given: Countable Choice, a real or complex Hilbert space $H$ , a compact self-adjoint $S$ , its set $Σ$ of nonzero eigenvalues, their eigenspaces $E_{t}$ , the orthogonal projections $P_{t}$ onto those spaces, and the orthogonal projection $P_{0}$ onto $\ker S$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the choice assumption in the compact self-adjoint spectral theorem; full Choice is not needed here.

[F1]

Spectral theorem for compact self adjoint operators: The nonzero eigenspaces are mutually orthogonal, $Σ$ is at most countable, and $H = \overset{}{span} {E_{t} : t \in Σ} \oplus \ker S$ .

[F2]

Orthogonal decomposition by a closed subspace: A closed subspace and its orthogonal complement give a unique orthogonal decomposition.

[F3]

Orthogonality and the orthogonal complement: Distinct vectors in orthogonal eigenspaces have zero inner product.

[F4]

The induced length is a norm: $∥ z ∥^{2} = ⟨ z, z ⟩$ .

[F5]

Real and complex inner product spaces, with the inner product linear in the first argument: The inner product is positive definite.

[F6]

The operator norm as the least bound and as the unit-sphere or unit-ball supremum: The operator norm is the supremum of $∥ T z ∥$ over $∥ z ∥ \leq 1$ .

[F7]

Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ : For a nonnegative family, its sum is the supremum of its finite subsums; if this sum is finite, tails outside a suitable finite set have arbitrarily small sum.

[F8]

The Cauchy-sequence reals have the least-upper-bound property: Every nonempty bounded-above set of real numbers has a supremum.

Proof

1.1A1F1F2F3F4

Write $x = P_{0} x + m$ with $m \in (\ker S)^{⊥}$ using [F1]–[F2]. For a finite $F \subseteq Σ$ , the vectors $P_{t} x$ are mutually orthogonal by [F1] and [F3]; their sum is the orthogonal projection of $m$ onto the finite orthogonal sum of the spaces $E_{t}$ . Pythagoras therefore gives $\sum_{t \in F} ∥ P_{t} x ∥^{2} \leq ∥ m ∥^{2} \leq ∥ x ∥^{2}$ .

2.1F1F2F4F7F8step 1.1algebra

The set ${∣ q (t) ∣ : t \in {0} \cup Σ}$ is nonempty and bounded, so [F8] gives its finite supremum $M_{q}$ . For finite $F \subseteq Σ$ , orthogonality and step 1.1 give $∥ \sum_{t \in F} q (t) P_{t} x ∥^{2} = \sum_{t \in F} ∣ q (t) ∣^{2} ∥ P_{t} x ∥^{2} \leq M_{q}^{2} ∥ x ∥^{2}$ . Put $a_{t} = ∥ P_{t} x ∥^{2}$ . Step 1.1 bounds every finite subsum of $(a_{t})$ by $∥ x ∥^{2}$ , so [F7] gives a finite sum with tails arbitrarily small. If $M_{q} = 0$ , every weighted spectral term vanishes. Otherwise, for any $ε > 0$ , choose a finite $F_{0} \subseteq Σ$ with $\sum_{t \notin F_{0}} a_{t} < (ε / M_{q})^{2}$ . For finite $F, H \supseteq F_{0}$ , orthogonality gives $∥ \sum_{t \in F} q (t) P_{t} x - \sum_{t \in H} q (t) P_{t} x ∥^{2} \leq M_{q}^{2} \sum_{t \in F △ H} a_{t} < ε^{2}$ . Thus the finite-subset net is Cauchy; completeness of $H$ makes it norm-convergent.

3.1F4F5F6F7step 2.1algebra∎

Finite partial sums are linear in $x$ , so taking their norm limits shows that $x \mapsto q (0) P_{0} x + \sum_{t \in Σ} q (t) P_{t} x$ is linear. Orthogonality and convergence of the finite-subset net give $∥ Q_{q} x ∥^{2} = ∣ q (0) ∣^{2} ∥ P_{0} x ∥^{2} + \sum_{t \in Σ} ∣ q (t) ∣^{2} ∥ P_{t} x ∥^{2} \leq M_{q}^{2} ∥ x ∥^{2}$ , with the nonnegative sum understood as in [F7]. Thus $Q_{q}$ is bounded and [F6] gives $∥ Q_{q} ∥ \leq M_{q}$ .

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Sources

Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 (standard reference, not scraped)