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Norm need not equal spectral radius

Statement refuted

Assume the Axiom of Choice (The Axiom of Choice). In $M_{2} (C)$ with the Euclidean operator norm the matrix

$E_{12} = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array})$

has operator norm $∥ E_{12} ∥ = 1$ and spectral radius $r (E_{12}) = 0$ (Spectral radius). So the norm of an element of a unital Banach algebra need not equal its spectral radius.

Facts & Assumptions

Given: The Axiom of Choice, the algebra $M_{2} (C)$ with the Euclidean operator norm, and the matrix $E_{12}$ acting on column vectors $(x, y) \in C^{2}$ .

[L1]

In $M_{2} (C)$ the spectrum of a matrix $A$ is ${λ : \det (λ I - A) = 0}$ and the algebra is a unital Banach algebra with the operator norm (Spectrum in a finite-dimensional matrix algebra).

[L2]

The spectral radius is $r (a) = \max {∣ z ∣ : z \in σ (a)}$ , and it is defined under the Axiom of Choice (Spectral radius, Spectrum and resolvent set in a Banach algebra).

Counterexample

technique · direct

1.1

$E_{12} (x, y) = (y, 0)$ , so $∥ E_{12} (x, y) ∥ = ∣ y ∣ \leq ∥ (x, y) ∥$ for every $(x, y) \in C^{2}$ (operator norm on the Euclidean plane), with equality at $(0, 1)$ ; hence $∥ E_{12} ∥ = 1$ .

algebra

2.1

$E_{12}^{2} = 0$ , and $\det (λ I - E_{12}) = λ^{2}$ , whose only zero is $λ = 0$ ; by [L1] the spectrum is $σ (E_{12}) = {0}$ .

step 1.1L1algebra

3.1

By [L2] the spectral radius is $r (E_{12}) = \max {∣ z ∣ : z \in {0}} = 0 < 1 = ∥ E_{12} ∥$ ; hence the two quantities differ for this element.

step 1.1step 2.1L2algebra∎

Remarks

The witness is a nonzero nilpotent of minimal size. $E_{12}$ is the smallest nonzero nilpotent: its square vanishes and its norm is one, so the gap between norm and spectral radius is already visible on the unit sphere of the matrix algebra.
The spectral radius formula records the same gap asymptotically. $∥ E_{12}^{n} ∥^{1 / n}$ equals $1$ for $n = 1$ and $0$ for $n \geq 2$ , and its limit is $0 = r (E_{12})$ , in agreement with Spectral radius formula.

Depends on

Used by

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Sources

Theo Bühler and Dietmar A. Salamon, Functional Analysis — §5.2.2, printed p. 222 (standard reference, not scraped)