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Weyl's theorem: invariance of the essential spectrum

Statement

Assume the Axiom of Choice. Let $A, C$ be self-adjoint operators such that $R_{A} (z) - R_{C} (z)$ is compact for one nonreal $z$ (equivalently, for every $z \in ρ (A) \cap ρ (C)$ ). Then $σ_{e s s} (A) = σ_{e s s} (C)$ . In particular a bounded self-adjoint compact perturbation preserves the essential spectrum, and if $K$ is symmetric and $A$ -compact, then $σ_{e s s} (A + K) = σ_{e s s} (A)$ with $D (A + K) = D (A)$ .

Facts & Assumptions

[A1]

$λ \in σ_{e s s} (S)$ exactly when $S$ has an orthonormal singular Weyl sequence at $λ$ (Weyl criterion for the essential spectrum, Discrete and essential spectrum of a self-adjoint operator).

[A2]

For $z \in ρ (S)$ and $λ \in R$ one has $R_{S} (z) + \frac{1}{λ - z} I = - \frac{1}{λ - z} R_{S} (z) (S - λ)$ on $D (S)$ , equivalently $R_{S} (z) (S - λ) = - (λ - z) (R_{S} (z) + \frac{1}{λ - z} I)$ (Resolvent and spectrum of an unbounded operator).

[A3]

A compact operator maps weakly convergent sequences to norm convergent sequences, and $R_{S} (z)$ is bounded (Compact operator sends weakly convergent sequences to norm convergent sequences, Compact linear operator).

[A4]

For self-adjoint $A, C$ and nonreal $z, w$ , the bounded resolvent identity gives $D_{w} = [I + (w - z) R_{A} (z)]^{- 1} D_{z} [I - (w - z) R_{C} (w)]$ , where $D_{z} : = R_{A} (z) - R_{C} (z)$ and the inverse first factor is $I + (z - w) R_{A} (w)$ . Hence compactness of $D_{z}$ transfers to $D_{w}$ , and conversely by exchanging $z, w$ (The resolvent star algebra is dense in C_0(R), Compositions with a compact operator are compact).

[A5]

An $A$ -compact symmetric $K$ has $A$ -bound zero, so Kato-Rellich makes $A + K$ self-adjoint on $D (A)$ ; the second resolvent identity $R_{A + K} (z) - R_{A} (z) = R_{A + K} (z) K R_{A} (z)$ holds for $z$ in the common resolvent set (Relative compactness with respect to an operator, Kato-Rellich theorem, Second resolvent identity for a closed perturbation).

Proof

technique · direct

Given: Self-adjoint $A, C$ with compact resolvent difference at a nonreal $z$ .

1.1

Let $λ \in σ_{e s s} (A)$ and let $(x_{n})$ be the orthonormal Weyl sequence of [A1]. By [A2] and $∥ (A - λ) x_{n} ∥ \to 0$ one has $(R_{A} (z) + \frac{1}{λ - z}) x_{n} \to 0$ ; since $R_{A} (z) - R_{C} (z)$ is compact and $x_{n} ⇀ 0$ , [A3] gives $(R_{C} (z) + \frac{1}{λ - z}) x_{n} \to 0$ as well.

A1A2A3

1.2

Parameter independence is [A4].

2.1

Then $(C - λ) R_{C} (z) x_{n} = (z - λ) R_{C} (z) x_{n} - x_{n} \to 0$ by [A2] and step 1.1, and $∥ R_{C} (z) x_{n} ∥ \to ∣ λ - z ∣^{- 1} > 0$ ; the normalized vectors $y_{n} : = ∥ R_{C} (z) x_{n} ∥^{- 1} R_{C} (z) x_{n}$ lie in $D (C)$ , have unit norm, converge weakly to $0$ and satisfy $(C - λ) y_{n} \to 0$ , so they form a singular Weyl sequence and $λ \in σ_{e s s} (C)$ by [A1]. Interchanging the roles of $A$ and $C$ gives equality.

A1A2step 1.1

3.1

Bounded compact perturbations: if $K$ is bounded, symmetric and compact, then $A + K$ is self-adjoint with $D (A + K) = D (A)$ by Kato-Rellich applied with the admissible pair $(0, ∥ K ∥)$ , and the second resolvent identity gives $R_{A + K} (z) - R_{A} (z) = R_{A + K} (z) K R_{A} (z)$ , compact as a product of the compact $K$ with bounded factors; so $σ_{e s s} (A + K) = σ_{e s s} (A)$ by step 2.1.

A4A5step 2.1

3.2

$A$ -compact perturbations: for symmetric $K$ that is $A$ -compact, [A5] makes $A + K$ self-adjoint with $D (A + K) = D (A)$ and gives $R_{A + K} (z) - R_{A} (z) = R_{A + K} (z) K R_{A} (z)$ , a product of the bounded operator $R_{A + K} (z)$ with the compact operator $K R_{A} (z)$ , hence compact; then step 2.1 applies.

A5step 2.1

4.1

The claims are steps 1.1, 1.2 and 2.1 (compact resolvent difference), 3.1 (bounded compact perturbations) and 3.2 ( $A$ -compact perturbations). ∎

Depends on

Used by

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Sources

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