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Spectral form domain and core of a semibounded operator

Statement

Assume the Axiom of Choice. Let $A$ be self-adjoint on a complex Hilbert space H, with $A \geq c I$ meaning $⟨ A x, x ⟩ \geq c ∥ x ∥^{2}$ for every $x \in D (A)$ , for some real $c$ . Put $Q (A) : = D ((A - c I)^{1 / 2}), q_{A} [x] : = c ∥ x ∥^{2} + ∥ (A - c I)^{1 / 2} x ∥^{2} (x \in Q (A)) .$ Then $Q (A)$ and $q_{A}$ do not depend on the choice of the constant $c \leq \inf σ (A)$ (the form $q_{A} [x] = \int λ d E_{x}$ is itself unchanged, while the summand $∥ (A - c I)^{1 / 2} x ∥^{2}$ changes by the constant $(c - c^{'}) ∥ x ∥^{2}$ when $c$ is replaced by $c^{'} \leq c$ ), $q_{A} [x] = \int λ d E_{x} (λ) = ⟨ A x, x ⟩$ for $x \in D (A)$ , the domain $D (A)$ is dense in $Q (A)$ for the norm $∥ x ∥_{Q} = (∥ x ∥^{2} + ∥ (A - c I)^{1 / 2} x ∥^{2})^{1 / 2}$ , and $q_{A}$ is a closed quadratic form with $q_{A} [x] \geq c ∥ x ∥^{2}$ .

Here the square root is the Borel calculus of $\sqrt{\max (λ - c, 0)}$ ; the proof shows that E is carried on $[c, \infty)$ . A closed semibounded quadratic form means the diagonal of a Hermitian sesquilinear form on a dense linear domain, complete in the displayed shifted form norm. If H={0}, use the unique PVM and operator and the convention inf(empty spectrum)=+infinity.

Facts & Assumptions

[A1]

The spectral theorem gives $D (A) = {x : \int λ^{2} d E_{x} < \infty}$ and A as the coordinate integral on nonzero H. The unbounded integral is closed, has linear domain, squared norm integral, and real-function pairing $⟨ f (E) x, x ⟩ = \int f d E_{x}$ . Real f gives a self-adjoint operator. The zero-space integral is defined directly. Spectral theorem for unbounded self-adjoint operators (PVM form) The unbounded PVM integral is densely defined, closed and normal Integral of a measurable function against a projection-valued measure

[A2]

Projections multiply by intersection, and $E_{x} (B) = ∥ E (B) x ∥^{2}$ is a finite measure of mass $∥ x ∥^{2}$ . Consequently $E_{E (J) x} (B) = E_{x} (B \cap J)$ , using $E (B) E (J) = E (B \cap J)$ and the norm formula; complementary projections give the analogous complementary restriction. The spectrum of A is the essential range of the coordinate function. Projection valued measure Unbounded Borel functional calculus: domains, products, spectral mapping

[A3]

Scalar dominated convergence applies to the finite measures E_x. Dominated convergence

[A4]

H is complete and its inner product is first-linear. Self-adjoint operators have dense linear domains. The assumed AC directly supplies every choice function required by the PVM, closed-integral and spectral-theorem interfaces. Hilbert space Symmetric, self-adjoint and essentially self-adjoint operators The Axiom of Choice

Proof

technique · direct

Given: AC, self-adjoint A and its lower bound c.

1.1

If H={0}, every domain and form consists of zero, all norms vanish, and every assertion follows directly from the zero-space convention in [A1]. Otherwise obtain E from [A1], under the choice assumption in [A4]. For $J_{m} = [- m, c - 1 / m]$ (empty intervals allowed), a vector $v = E (J_{m}) x$ belongs to D(A) by [A2] and boundedness of lambda on J_m. If v were nonzero then $⟨ A v, v ⟩ = \int_{J_{m}} λ d E_{v} \leq (c - 1 / m) ∥ v ∥^{2}$ , contradicting the lower bound. Thus E(J_m)=0 for all positive integers m. Their union is $(- \infty, c)$ , whose scalar measures therefore vanish by countable subadditivity. The projection norm formula gives E((-infinity,c))=0. The essential-range description in [A2] implies $σ (A) \subseteq [c, \infty)$ .

A1A2A4given

2.1

Put $B = \sqrt{\max (λ - c, 0)} (E)$ . It is closed and self-adjoint by [A1], and step 1.1 gives $D (B) = {x : \int (λ - c) d E_{x} < \infty}$ and $∥ B x ∥^{2} = \int (λ - c) d E_{x}$ . Integrals here and below can be restricted to [c,infinity). Since $∣ λ ∣ \leq (λ - c) + ∣ c ∣$ there, lambda is absolutely integrable for x in Q(A). Hence $q_{A} [x] = c ∥ x ∥^{2} + ∥ B x ∥^{2} = \int λ d E_{x}$ and $q_{A} [x] \geq c ∥ x ∥^{2}$ .

A1A2step 1.1

3.1

For any other lower spectral bound c'<=c, $λ - c^{'} = (λ - c) + (c - c^{'})$ on the carrier. Since E_x has finite mass, the two domain integrals are finite simultaneously. Adding the appropriate constant times the mass gives the same q_A, while the square-root squared norm increases by $(c - c^{'}) ∥ x ∥^{2}$ . Two arbitrary admissible lower bounds can be compared in their numerical order, so this proves full independence. Their squared form norms differ by that same multiple of $∥ x ∥^{2}$ , hence are equivalent since each dominates $∥ x ∥^{2}$ .

A2step 2.1

3.2

If x belongs to D(A), then $λ - c \leq 1 + λ^{2} + ∣ c ∣$ on the carrier, so x belongs to Q(A). By [A1] and step 2.1, $q_{A} [x] = \int λ d E_{x} = ⟨ A x, x ⟩$ .

A1A2step 2.1

3.3

For x in Q(A), set $x_{n} = E ([- n, n]) x$ , n>=1. By [A2], $\int λ^{2} d E_{x_{n}} \leq n^{2} ∥ x ∥^{2}$ , so x_n belongs to D(A). The same restriction identity gives $∥ x - x_{n} ∥_{Q}^{2} = \int_{∣ λ ∣ > n} (1 + λ - c) d E_{x} \to 0$ by dominated convergence, with nonnegative integrable majorant $1 + λ - c$ on the carrier. This is the asserted form-norm density, with the exact identity $∥ z ∥_{Q}^{2} = q_{A} [z] + (1 - c) ∥ z ∥^{2}$ .

A1A2A3step 2.1

4.1

The form is the diagonal of $a (x, y) = c ⟨ x, y ⟩ + ⟨ B x, B y ⟩$ on the linear domain D(B); this is Hermitian and sesquilinear by [A4]. Its domain is dense in H because it contains D(A) by step 3.2. For a Cauchy sequence in the form norm, both x_n and Bx_n are Cauchy in H. Completeness gives limits x and y. Closedness of B implies x in D(B) and Bx=y. Therefore $∥ x_{n} - x ∥_{Q}^{2} = ∥ x_{n} - x ∥^{2} + ∥ B x_{n} - B x ∥^{2} \to 0$ , proving completeness and closedness in the stated sense.

A1A4step 2.1step 3.2

5.1

Steps 2.1 and 3.1 establish the domain, integral identity, lower bound and independence; steps 3.2 and 3.3 give the operator-domain identity and core, and step 4.1 gives the closed quadratic form. Positive integer cutoffs are specified without choices. AC is inherited through [A4]; negative and zero lower bounds are allowed without taking a square root of q_A itself. The zero Hilbert space was handled in step 1.1.

A4step 1.1step 2.1step 3.1step 3.2step 3.3step 4.1∎

Depends on

Used by

Min-max principle below the essential spectrum Theorem

Dependency tree · two levels

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Sources

Gerald Teschl, Mathematical Methods in Quantum Mechanics, second edition (standard reference, not scraped)