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Self adjoint norm and spectrum extrema

Statement

Assume AC. If $T \in B (H)$ is bounded and self-adjoint on a nonzero complex Hilbert space $H$ , then $\min σ (T) I \leq T \leq \max σ (T) I$ , these bounds are sharp, and $∥ T ∥ = \max {∣ \min σ (T) ∣, ∣ \max σ (T) ∣}$ .

Facts & Assumptions

[A1]

For self-adjoint $T$ the calculus sends $z$ to $T$ , and continuous composition and positivity hold: $f (T)$ is positive whenever $f \geq 0$ , and $∥ f (T) ∥ = ∥ f ∥_{\infty}$ (Continuous functional calculus for bounded self adjoint operators, Continuous functional calculus properties).

[A2]

$S \leq R$ means $⟨ (R - S) x, x ⟩ \geq 0$ for every $x$ , an order on the real vector space of bounded self-adjoint operators; $S \geq 0$ is positivity (Order on bounded self adjoint operators, Self-adjoint, positive, unitary and normal operators).

[A3]

A bounded positive operator has spectrum in $[0, + \infty)$ (Spectrum of a positive operator is nonnegative).

[A4]

The operator spectrum is the spectrum in the nonzero unital Banach algebra $B (H)$ , since invertibility there means exactly a bounded two-sided operator inverse. It is nonempty and compact, and is real for self-adjoint $T$ (Spectrum and resolvent of a bounded operator, Bounded Hilbert operators form a C star algebra, Spectrum is nonempty compact and norm bounded, Spectrum of a self adjoint operator is real). The identity real function on this compact set attains its minimum and maximum (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[A5]

Continuous functions on $σ (T)$ satisfy the pointwise identities used below: $z - m \geq 0$ if $m \leq \min σ (T)$ , and $M - z \geq 0$ if $M \geq \max σ (T)$ (Self-adjoint, positive, unitary and normal operators for the scalar-multiple convention used in the calculus).

[A6]

AC is the hypothesis of the calculus and spectral suppliers (The Axiom of Choice).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ and a bounded self-adjoint $T \in B (H)$ , with $m : = \min σ (T)$ and $M : = \max σ (T)$ .

1.1

The spectrum is a nonempty compact subset of $R$ , so $m$ and $M$ exist and $σ (T) \subseteq [m, M]$ .

1.2

The functions $z - m$ and $M - z$ are continuous and nonnegative on $σ (T)$ , so $(z - m) (T) = T - m I$ and $(M - z) (T) = M I - T$ are positive operators.

A1A5

1.3

The norm identity: $∥ T ∥ = ∥ z (T) ∥ = ∥ z ∥_{\infty, σ (T)} = \max_{λ \in σ (T)} ∣ λ ∣ = \max (∣ m ∣, ∣ M ∣)$ , since $σ (T) \subseteq [m, M]$ .

2.1

Consequently $m I \leq T \leq M I$ in the order of the definition, since the two differences are positive operators.

step 1.2A2

2.2

Sharpness of the lower bound: if $c I \leq T$ for a real $c$ , then $T - c I$ is positive, so its spectrum lies in $[0, + \infty)$ ; and $σ (T - c I) = σ (T) - c$ , because $λ I - (T - c I) = (λ + c) I - T$ has exactly the same bounded-invertibility condition. This spectrum contains $m - c$ , hence $m - c \geq 0$ and $c \leq m$ . If $T \leq d I$ , then $d I - T$ is positive and $σ (d I - T) = d - σ (T)$ : $λ I - (d I - T) = - ((d - λ) I - T)$ is boundedly invertible exactly when $(d - λ) I - T$ is. Thus $d - M \geq 0$ .

step 1.1A2A3A4algebra

3.1

Sharpness of both bounds: $m I \leq T \leq M I$ by step 2.1, and any lower bound is at most $m$ while any upper bound is at least $M$ by step 2.2, so $m$ and $M$ are the greatest lower bound and least upper bound of the quadratic form on unit vectors.

step 2.1step 2.2

4.1

Therefore $\min σ (T) I \leq T \leq \max σ (T) I$ with sharp bounds, and $∥ T ∥ = \max {∣ \min σ (T) ∣, ∣ \max σ (T) ∣}$ .

step 1.3step 3.1A6∎

Depends on

Used by

Spectral projections and resolution of the identity Corollary

Dependency tree · two levels

69 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, Lemma 5.49 and §5.4, printed pp.238–255 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem, §4, pp.10–15 (standard reference, not scraped)