Lecture 2. Quadratic forms in Hilbert spaces

Quadratic forms and generators
Semigroups and generators
A first example: The Dirichlet energy on an open set

Quadratic forms and generators

Definition A quadratic form ℰ on H is a non-negative definite, symmetric bilinear form 𝒟(ℰ) × 𝒟(ℰ) → ℝ, where 𝒟(ℰ) is a dense subspace of H. A quadratic form ℰ on H is said to be closed if 𝒟(ℰ) equipped with the norm

‖f‖_𝒟(ℰ)²= ‖f‖² + ℰ(f,f)

is a Hilbert space. A quadratic form ℰ on H is said to be closable it admits a closed extension, i.e. there exists a closed quadratic form ℰ’ such that 𝒟(ℰ) ⊂ 𝒟(ℰ’) and ℰ’ coincides with ℰ on 𝒟(ℰ) × 𝒟(ℰ).

Lemma A quadratic form ℰ is closable if and only if for any sequence f_n in 𝒟(ℰ) such that f_n → 0 in H and ℰ(f_n – f_m, f_n – f_m) → 0 when n, m → +∞ one has ℰ(f_n, f_n) → 0.

Proof

On 𝒟(ℰ), let us consider the following norm

||f||_ℰ² = ||f||² + ℰ(f, f).

By completing 𝒟(ℰ) with respect to this norm, we get an abstract Hilbert space (H_ℰ,⟨⋅,⋅⟩_ℰ). Since for f ∈ 𝒟(ℰ), ||f|| ≤ ||f||_ℰ, the injection map ι : (𝒟(ℰ), || · ||_ℰ) → (H, || · ||) is continuous and it may therefore be extended into a continuous map ῑ : (H_ℰ, || · ||_ℰ) → (H, || · ||). Let us show that ̄ι is injective so that H_ℰ may be identified with a subspace of H. So, let f ∈ H_ℰ such that ῑ(f) = 0. We can find a sequence f_n ∈ 𝒟(ℰ), such that || f_n – f ||_ℰ → 0 and || f_n || → 0. We have then

||f||_ℰ² = lim_{n → + ∞} ⟨f_n, f_n⟩_ℰ

= lim_{n → + ∞} ⟨f_n,f_n⟩ + ℰ(f_n, f_n)

= 0,

thus f=0 and ῑ is injective. Therefore, H_ℰ may be identified with a subspace of H and the quadratic form on H defined by

ℰ'(f) = ||f||_ℰ² – ||f||², f ∈ H_ℰ

is closed because (H_ℰ,⟨⋅,⋅⟩_ℰ) is a Hilbert space and obviously is an extension of ℰ.

If a quadratic form ℰ is closable, then its minimal closed extension is called the closure of ℰ. In that case, one can easily check that the closure of ℰ is actually the quadratic form ℰ constructed in the previous proof.

Theorem Let ℰ be a closed symmetric non-negative bilinear form on H. There exists a unique densely defined non-positive self-adjoint operator A on H defined by

𝒟(A) = {f ∈ H, ∃g ∈ H, ∀h ∈ H, ℰ(f,h) = -⟨h,g⟩}

Af = g.

The operator A is called the generator of ℰ. Conversely, if A is a densely defined non-positive self-adjoint operator on H, one can define a closed symmetric non-negative bilinear form ℰ on H by

𝒟(ℰ) = 𝒟((-A)^1/2), ℰ(f,g) = ⟨(-A)^1/2f,(-A)^1/2g⟩.

Proof

Let ℰ be a closed symmetric non-negative bilinear form on H. As usual, we denote by ℱ the domain of ℰ. We note that for λ > 0, ℱ equipped with the norm (||f||² + λℰ(f))^1/2 is a Hilbert space because ℰ is closed. From the Riesz representation theorem, there exists then a linear operator R_λ :H → ℱ such that for every f ∈ H, g ∈ ℱ

⟨f,g⟩ = λ⟨R_λ f , g⟩ + ℰ(R_λ f,g).

From the definition, the following properties are then easily checked:

||R_λf|| ≤ (1/λ) ||f|| (apply the definition of R_λ with g = R_λf and then use the Cauchy-Schwarz inequality);
For every f,g ∈ H, ⟨R_λf , g⟩ = ⟨f , R_λg⟩;
R_λ₁ – R_λ₂ + (λ₁ – λ₂)R_λ₁R_λ₂ = 0;
For every f ∈ H, lim_{λ → +∞} || λR_λf -f || = 0.

We then claim that R_λ is invertible. Indeed, if R_λf = 0, then for α > λ , one has from 3, R_αf = 0. Therefore f = lim_{α → +∞} R_αf = 0. Denote then

Af = λf – R_λ^-1f,

and 𝒟(A) is the range of R_λ. It is straightforward to check that A does not depend on λ. The operator A is a densely defined self-adjoint operator that satisfies the properties stated in the theorem (Exercise !).

Conversely, if A is a densely defined non-positive self-adjoint operator on H, then (-A)^1/2 is a densely defined self-adjoint operator and the quadratic form

ℰ(f,g) := ⟨(-A)^1/2f, (-A)^1/2g⟩

is closed and densely defined on 𝒟((-A)^1/2).

Exercise Prove the properties 1,2,3,4 of the previous proof.

In practice, the following lemma is often useful to construct closed quadratic forms and easily follows from the previous results.

Lemma Let A be a densely defined non-positive symmetric operator 𝒟(A) → H. The quadratic form

ℰ(f,g) = -⟨f, Ag⟩, f,g ∈ 𝒟(A)

is closable and the generator of its closure is a self-adjoint extension of A.

Semigroups and quadratic forms

Theorem Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction semigroup on H. One can define a closed quadratic form on H by

ℰ(f,f) := lim_{t → 0} ⟨(Id – P_t)/t f, f⟩,

where the domain of this form is the set of f‘s for which the limit exists. The quadratic form ℰ is called the quadratic form associated to the semigroup (P_t)_t≥0.

Proof

Let A be the generator of the semigroup (P_t)_{t ≥ 0}. We use spectral theorem to represent A as

U^-1 A U g(x) = -λ(x) g(x),

so that

U^-1 P_t U g(x) = e^-tλ(x) g(x).

We then note that for every g ∈ L²(Ω,ν),

⟨(Id – P_t)/t Ug, Ug⟩ = ∫_Ω (1 – e^-tλ(x))/t g(x)² dν(x).

This proves that for every f ∈ H, the map t → ⟨(Id – P_t)/t f, f⟩ is non-increasing. Therefore, the limit lim_{t → 0} ⟨(Id – P_t)/t f, f⟩ exists if and only if ∫_Ω (U^-1f)²(x) λ(x) dν(x) < +∞, which is equivalent to the fact that f ∈ 𝒟((-A)^1/2). In which case we have

lim_{t → 0} ⟨(Id – P_t)/t f, f⟩ = ||(-A)^1/2f||².

Since (-A)^1/2 is a densely defined self-adjoint operator, the quadratic form

ℰ(f) := ||(-A)^1/2f||²

is closed and densely defined on ℱ := 𝒟((-A)^1/2).

A first example: The Dirichlet energy on an open set Ω ⊂ ℝⁿ

Let Ω ⊂ ℝⁿ be an open connected set. We do not assume any regularity on the boundary of Ω. Classically, one can define the (1,2) Sobolev space

W^1,2(Ω) = {f ∈ L²(Ω) : ∂f/∂x_i ∈ L²(Ω)}

where the derivatives ∂u/∂x_i are understood in the weak sense. The quadratic form

ℰ(f,g) = ∫_Ω ⟨∇f, ∇g⟩ dx = ∑_i=1ⁿ ∫_Ω ∂f/∂x_i ∂g/∂x_i dx

with domain W^1,2(Ω) is then a closed densely defined quadratic form on L²(Ω) since it is well-known that the Sobolev norm

||f||²_W^1,2(Ω) = ||f||²_L²(Ω) + ||∇f||²_L²(Ω)

is complete. The generator of the form ℰ is called the Neumann Laplacian on Ω.

On the other hand, let

Δ = ∑_i=1ⁿ ∂²/∂x_i²

be the usual Laplacian on ℝⁿ, the derivatives being understood in the ordinary sense, and C_c^∞(Ω) be the set of smooth functions with a compact support included in Ω. Then, from a lemma, the quadratic form

ℰ₀(f,g) = -∫_Ω f Δg dx

with domain C_c^∞(Ω) is closable. The domain of the closure of ℰ₀ is the Sobolev space W₀^1,2(Ω) and the generator of the closure of ℰ₀ is called the Dirichlet Laplacian on Ω.

Notice that both the Neumann and the Dirichlet Laplacian are self-adjoint extensions of the Laplacian Δ with domain C_c^∞(Ω). In general, the Neumann and Dirichlet Laplacian do not coincide. For instance if the boundary of Ω is smooth, then smooth functions in the domain of the Neumann Laplacian have vanishing normal derivatives while smooth functions in the domain of the Dirichlet Laplacian vanish on the boundary of u.