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Spectrum of a compact operator is countable with only zero as possible accumulation

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a complex Banach space, let $K : X \to X$ be a compact operator (Compact linear operator) and let $σ (K)$ be its spectrum (Spectrum and resolvent of a bounded operator). Then:

$σ (K)$ is at most countable (Finite, countably infinite, countable, uncountable);
for every $λ \in C$ with $λ \neq 0$ there is a real $r > 0$ with $B (λ, r) \cap σ (K) \subseteq {λ}$ (Open ball, closed ball and sphere in a metric space).

In particular the only point of $C$ that can be an accumulation point of $σ (K)$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space) is $0$ .

Facts & Assumptions

[A1]

Under AC, for every real $ε > 0$ the set $S_{ε} : = {λ \in σ (K) : ∣ λ ∣ \geq ε}$ is finite (Riesz schauder spectrum of a compact operator).

[A2]

For every real $ε > 0$ there is a natural $n \geq 1$ with $1 / n < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[A3]

Under ${A C}_{ω}$ , an at most countable union of at most countable sets is at most countable (Countable unions of at most countable sets, assuming ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ ), Finite, countably infinite, countable, uncountable), and every subset of an at most countable set is at most countable (Every subset of an at most countable set is at most countable); $A C$ implies ${A C}_{ω}$ (AC supplies the countable and dependent choices used in Banach integration, The Axiom of Choice).

[A4]

In $C$ the balls are those of the metric $d (z, w) = ∣ z - w ∣$ (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane, Open ball, closed ball and sphere in a metric space); a point $λ$ is an accumulation point of a set $A$ when every punctured ball $B (λ, r) ∖ {λ}$ meets $A$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

Proof

technique · direct

Given: $A C$ , a complex Banach space $X$ , a compact $K : X \to X$ , and the sets $S_{1 / n} = {λ \in σ (K) : ∣ λ ∣ \geq 1 / n}$ , $n \geq 1$ .

1.1

Every subspace of $σ (K)$ of the form $S_{1 / n}$ is finite by [A1]; and every $μ \in σ (K)$ with $μ \neq 0$ lies in some $S_{1 / n}$ , because $∣ μ ∣ > 0$ and [A2] gives $n$ with $1 / n < ∣ μ ∣$ .

A1A2

1.2

The set $U : = {0} \cup ⋃_{n \geq 1} S_{1 / n}$ is at most countable: it is a countable union of finite sets, and [A3] applies.

A1A3

2.1

$σ (K) \subseteq U$ by [step 1.1].

step 1.1

2.2

Claim 2: let $λ \neq 0$ and choose $n$ with $1 / n \leq ∣ λ ∣ / 2$ by [A2]. Every $z \in B (λ, ∣ λ ∣ / 2)$ satisfies $∣ z ∣ \geq ∣ λ ∣ - ∣ z - λ ∣ > ∣ λ ∣ / 2 \geq 1 / n$ , so the set $F : = σ (K) \cap B (λ, ∣ λ ∣ / 2)$ is contained in $S_{1 / n}$ and is finite by [step 1.1]. Put $E : = F ∖ {λ}$ . If $E$ is nonempty, the finite set of positive numbers ${∣ z - λ ∣ : z \in E}$ has a minimum $ρ > 0$ ; if $E$ is empty, put $ρ : = ∣ λ ∣ / 2$ . For $r : = \min (ρ, ∣ λ ∣ / 2) > 0$ , any $z \in B (λ, r) \cap σ (K)$ lies in $F$ , while $z \neq λ$ would put $z$ in $E$ and give the contradiction $∣ z - λ ∣ \geq ρ \geq r > ∣ z - λ ∣$ . Hence $B (λ, r) \cap σ (K) \subseteq {λ}$ .

step 1.1A2A4

3.1

Claim 1: $σ (K)$ is at most countable, being a subset of the at most countable set $U$ , by [A3]; moreover $U$ is at most countable by [step 1.2] and $σ (K) \subseteq U$ by [step 2.1].

step 1.2step 2.1A3

4.1

Finally, if $λ \neq 0$ and every punctured ball around $λ$ met $σ (K)$ , then by [step 2.2] the punctured ball $B (λ, r) ∖ {λ}$ meets $σ (K)$ yet contains none of its points, a contradiction; so the only possible accumulation point is $0$ , and claims 1 and 2 are [step 3.1] and [step 2.2].

step 3.1step 2.2A4∎

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Sources

Theo Bühler and Dietmar Salamon, Functional Analysis — §5.2.3 p.225, Theorem 5.21 and the countability remark (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.6, spectrum of a compact operator (standard reference, not scraped)