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Finite dimensionality of invisible spaces

PDF p. 11: Lemma 3.4 proof (i)

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , the nonempty compact obstacle $Θ \subset R^{n}$ has smooth boundary, its exterior $Ω = R^{n} ∖ Θ$ is connected, and $Θ$ is non-trapping in the generalized-ray sense. Fix $r > 0$ with $Θ \subset B_{r} (0)$ and an admissible pair of real cutoffs: $φ \in C^{\infty} (R^{n}; [0, 1])$ vanishes for $∣ x ∣ \leq r$ , is positive for $∣ x ∣ > r$ , and equals one for $∣ x ∣ \geq r + 1$ ; $ψ \in C_{c}^{\infty} (R^{n}; [0, 1])$ equals one on $B_{r + 2} (0)$ and has support in $B_{r + 3} (0)$ . Use this fixed $φ$ in the invisible space $N_{T}^{σ}$ and this fixed $ψ$ in the remainder $Q_{σ}$ throughout. Let $T > 0$ and $σ \in [- 2, 2)$ . The closed unit ball of the invisible space $N_{T}^{σ} \subset H_{D}^{σ} (Ω)$ is compact in the norm of $H_{D}^{σ} (Ω)$ , and $N_{T}^{σ}$ is finite-dimensional.

Statement references

def-se-nontrapping

Control region and nested real cutoffs

Invisible initial-data spaces

Regularized lower-order observation remainder

Open cover, subcover, compact metric space, and compact subset of a metric space

Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis

Facts & Assumptions

[F1]

The scale $H_{D}^{σ} (Ω)$ is the completion in the norm induced by the shifted Dirichlet Laplacian; it is a complex Hilbert space, with an onto isometric identification with the underlying $L^{2}$ Hilbert space supplied by the spectral-scale construction. Dirichlet spectral Sobolev scale and duality Positive spectral Hilbert scale: duality and interpolation

[F2]

With $W (t) = e^{i t Δ_{D}}$ and the fixed real observation cutoff $φ$ , $N_{T}^{σ}$ consists of those $v_{0} \in H_{D}^{σ}$ for which $φ W (t) v_{0} = 0$ for every $t \in [0, T]$ . It is a closed complex linear subspace. Control region and nested real cutoffs Invisible initial-data spaces

[F3]

The nonnegative remainder $Q_{σ}$ is the squared time-space norm defined in Regularized lower-order observation remainder, using the fixed $ψ$ of the Statement. Control region and nested real cutoffs Under the stated geometry and Choice, if $x_{j} ⇀ x$ weakly in $H_{D}^{σ}$ , then $Q_{σ} (x_{j} - x) \to 0$ . Weak convergence means convergence under every bounded scalar-linear functional. Lemma 3.2: compactness of the remainder Weak convergence notation in normed state spaces

[F4]

Under the same hypotheses and for this fixed pair $φ, ψ$ , the observation estimate is $∥ x ∥_{H_{D}^{σ}}^{2} \leq C (\int_{0}^{T} ∥ φ W (t) x ∥_{H_{D}^{σ}}^{2} d t + Q_{σ} (x))$ for all $x \in H_{D}^{σ}$ , with $C$ independent of $x$ . Control region and nested real cutoffs Lemma 3.3: observability modulo a compact remainder

[F5]

Under Countable Choice every complex Hilbert space is reflexive; under the ultrafilter lemma, Dependent Choice and Hahn–Banach, every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. The assumed Axiom of Choice supplies these principles. Hilbert spaces are reflexive Reflexivity is equivalent to weak subsequential compactness of bounded sequences The Axiom of Choice

[F6]

Under Countable Choice and Dependent Choice, sequential compactness and compactness are equivalent for metric spaces. An infinite-dimensional normed space has a noncompact closed unit ball. For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice In an infinite-dimensional normed space the closed unit ball is not compact Open cover, subcover, compact metric space, and compact subset of a metric space Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis

Proof

[F1] The scale $H_{D}^{σ} (Ω)$ is the completion in the norm induced by the shifted Dirichlet Laplacian; it is a complex Hilbert space, with an onto isometric identification with the underlying $L^{2}$ Hilbert space supplied by the spectral-scale construction. Dirichlet spectral Sobolev scale and duality Positive spectral Hilbert scale: duality and interpolation

[F2] With $W (t) = e^{i t Δ_{D}}$ and the fixed real observation cutoff $φ$ , $N_{T}^{σ}$ consists of those $v_{0} \in H_{D}^{σ}$ for which $φ W (t) v_{0} = 0$ for every $t \in [0, T]$ . It is a closed complex linear subspace. Control region and nested real cutoffs Invisible initial-data spaces

[F3] The nonnegative remainder $Q_{σ}$ is the squared time-space norm defined in Regularized lower-order observation remainder, using the fixed $ψ$ of the Statement. Control region and nested real cutoffs Under the stated geometry and Choice, if $x_{j} ⇀ x$ weakly in $H_{D}^{σ}$ , then $Q_{σ} (x_{j} - x) \to 0$ . Weak convergence means convergence under every bounded scalar-linear functional. Lemma 3.2: compactness of the remainder Weak convergence notation in normed state spaces

[F4] Under the same hypotheses and for this fixed pair $φ, ψ$ , the observation estimate is $∥ x ∥_{H_{D}^{σ}}^{2} \leq C (\int_{0}^{T} ∥ φ W (t) x ∥_{H_{D}^{σ}}^{2} d t + Q_{σ} (x))$ for all $x \in H_{D}^{σ}$ , with $C$ independent of $x$ . Control region and nested real cutoffs Lemma 3.3: observability modulo a compact remainder

[F5] Under Countable Choice every complex Hilbert space is reflexive; under the ultrafilter lemma, Dependent Choice and Hahn–Banach, every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. The assumed Axiom of Choice supplies these principles. Hilbert spaces are reflexive Reflexivity is equivalent to weak subsequential compactness of bounded sequences The Axiom of Choice

[F6] Under Countable Choice and Dependent Choice, sequential compactness and compactness are equivalent for metric spaces. An infinite-dimensional normed space has a noncompact closed unit ball. For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice In an infinite-dimensional normed space the closed unit ball is not compact Open cover, subcover, compact metric space, and compact subset of a metric space Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis

Given: Choice, the geometry (H), the fixed admissible pair $φ, ψ$ , $T > 0$ , and $σ \in [- 2, 2)$ as in the Statement.

Goal: Prove compactness of the closed unit ball of $N_{T}^{σ}$ and finite dimensionality of $N_{T}^{σ}$ .

1.1F1F5given

Let $(x_{j})$ be any sequence in that unit ball. It is bounded in the ambient Hilbert space $H_{D}^{σ}$ . By reflexivity and weak subsequence compactness, after passing to a subsequence it converges weakly to some $x \in H_{D}^{σ}$ . The difference sequence is weakly null.

2.1F2F3F4step 1.1

For any two indices $j, k$ in this subsequence, linearity of $N_{T}^{σ}$ makes $x_{j} - x_{k}$ invisible. The observation term in F4 uses the same fixed $φ$ as $N_{T}^{σ}$ in F2 and therefore vanishes, giving $∥ x_{j} - x_{k} ∥_{H_{D}^{σ}}^{2} \leq C Q_{σ} (x_{j} - x_{k})$ . Since $Q_{σ}$ is a squared norm of a linear image, its triangle inequality gives $Q_{σ} (x_{j} - x_{k})^{1 / 2} \leq Q_{σ} (x_{j} - x)^{1 / 2} + Q_{σ} (x_{k} - x)^{1 / 2}$ . Both terms tend to zero by weak convergence and the compact-remainder lemma. Thus the subsequence is norm Cauchy.

3.1F1F2step 2.1

Completeness of $H_{D}^{σ}$ gives a norm limit $y$ . Norm convergence implies weak convergence by boundedness of every dual functional, so uniqueness of scalar limits gives $y = x$ . Since $N_{T}^{σ}$ is norm closed and every $x_{j}$ has norm at most one, $x \in N_{T}^{σ}$ and $∥ x ∥_{H_{D}^{σ}} \leq 1$ . Hence every sequence in the closed unit ball has a norm-convergent subsequence with limit in that ball.

4.1F6step 3.1∎

The norm induces a metric on the unit ball, so its sequential compactness implies compactness. With the inherited norm, $N_{T}^{σ}$ is a normed space and this is its closed unit ball. The noncompactness criterion for an infinite-dimensional normed space therefore implies that $N_{T}^{σ}$ is finite-dimensional.

Source notes

The original article is cited for attribution of the finite-dimensionality argument in the proof of Lemma 3.4, PDF page 11. The proof above uses the stated mathematical suppliers; it does not use the article's assertion as a premise.

Depends on

Used by · 1

Locally constant invisible spaces are Laplacian invariant

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2