CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Normal operator with zero spectrum is zero

Statement

Assume AC. A bounded normal operator whose spectrum is ${0}$ is the zero operator.

Facts & Assumptions

[A1]

For a bounded normal operator $T$ on a nonzero complex Hilbert space, $∥ T ∥ = r (T) = \max {∣ λ ∣ : λ \in σ (T)}$ (Normal operator norm equals spectral radius).

[A2]

The operator norm is the least bound of $T$ , so $∥ T ∥ = 0$ forces $T x = 0$ for every $x$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[A3]

The spectrum is a subset of $C$ ; a normal operator is one with $T^{*} T = T T^{*}$ , and the zero operator is normal (Spectrum and resolvent of a bounded operator, Self-adjoint, positive, unitary and normal operators).

[A4]

AC is the hypothesis of the norm-and-spectral-radius supplier (The Axiom of Choice).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ and a bounded normal $T \in B (H)$ with $σ (T) = {0}$ .

1.1

The spectral radius is $r (T) = \max {∣ λ ∣ : λ \in σ (T)} = ∣ 0 ∣ = 0$ .

2.1

Hence $∥ T ∥ = r (T) = 0$ by the spectral-radius identity for normal operators.

step 1.1A1A4

3.1

Since $∥ T ∥ = 0$ is a bound for $T$ , $∥ T x ∥ \leq 0 ∥ x ∥ = 0$ for every $x$ , so $T x = 0$ for every $x$ and $T = 0$ .

step 2.1A2∎

Depends on

Used by

A quasinilpotent operator need not be zero Counterexample

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Theo Bühler and Dietmar Salamon, Functional Analysis, §5.4, printed pp.245–255 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, §4, pp.10–13 (standard reference, not scraped)