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Norm of a self adjoint operator from its quadratic form

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a real or complex Hilbert space (Hilbert space) and let $T \in B (H)$ be a bounded self-adjoint operator (Self-adjoint, positive, unitary and normal operators, A bounded linear operator between normed spaces), with operator norm $∥ T ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum). Put $q (x) : = ⟨ T x, x ⟩$ for $x \in H$ and

$M : = \sup {∣ q (x) ∣ : ∥ x ∥ = 1},$

with the convention that a supremum over the empty set of reals is $0$ ; the empty case occurs only for $H = {0}$ , where the only operator is $T = 0$ . Then

$∥ T ∥ = M = \sup_{∥ x ∥ = 1} ∣ ⟨ T x, x ⟩ ∣ .$

The identity holds over both scalar fields, and in the case $H = {0}$ both sides equal $0$ .

Facts & Assumptions

Given: A real or complex Hilbert space $H$ , a bounded self-adjoint operator $T$ , the quadratic form $q (x) = ⟨ T x, x ⟩$ , and $M = \sup {∣ q (x) ∣ : ∥ x ∥ = 1}$ with the empty-supremum convention.

[A1]

Self-adjointness is the identity $⟨ T x, y ⟩ = ⟨ x, T y ⟩$ for all $x, y$ : a self-adjoint operator satisfies $T^{*} = T$ , and the Hilbert adjoint is characterised by $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ (Self-adjoint, positive, unitary and normal operators, The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities).

[A2]

Inner-product algebra. The pairing is linear in the first argument, conjugate-linear in the second, conjugate symmetric, and positive definite, and $∥ v ∥^{2} = ⟨ v, v ⟩$ (Real and complex inner-product spaces and their induced length). Consequently for real $t > 0$ and $x, y \in H$ one has $⟨ T (t x), t^{- 1} y ⟩ = t \cdot t^{- 1} ⟨ T x, y ⟩ = ⟨ T x, y ⟩$ , the expansion $q (x \pm u) = q (x) \pm ⟨ T x, u ⟩ \pm ⟨ T u, x ⟩ + q (u)$ holds, and if $⟨ T x, u ⟩$ is real then self-adjointness gives $⟨ T u, x ⟩ = \overline{⟨ T x, u ⟩} = ⟨ T x, u ⟩$ , so that $q (x + u) - q (x - u) = 4 ⟨ T x, u ⟩$ . For $z \neq 0$ with unit vector $u = z / ∥ z ∥$ one has $q (z) = ∥ z ∥^{2} q (u)$ .

[A3]

Parallelogram law. $∥ x + u ∥^{2} + ∥ x - u ∥^{2} = 2 ∥ x ∥^{2} + 2 ∥ u ∥^{2}$ for all $x, u \in H$ (Pythagoras, the parallelogram identity, and the real and complex polarisation identities).

[A4]

Operator norm and Cauchy–Schwarz. $∥ T v ∥ \leq ∥ T ∥ ∥ v ∥$ and $∥ T ∥ = \sup {∥ T v ∥ : ∥ v ∥ \leq 1}$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces); $∣ ⟨ v, w ⟩ ∣ \leq ∥ v ∥ ∥ w ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs), so the dual norm formula $∥ z ∥ = \sup {∣ ⟨ z, y ⟩ ∣ : ∥ y ∥ = 1}$ holds for every $z \in H$ by Cauchy–Schwarz and by testing $y = z / ∥ z ∥$ when $z \neq 0$ (both sides are $0$ at $z = 0$ ).

[A5]

Choice. Countable Choice is the hypothesis under which this pair's Hilbert-space interface is stated; no choice is used inside the argument below (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: Countable Choice, a real or complex Hilbert space $H$ , a bounded self-adjoint $T$ , and $M = \sup {∣ q (x) ∣ : ∥ x ∥ = 1}$ .

1.1

The quadratic form is dominated by $M$ . For $z \in H$ , if $z = 0$ then $∣ q (z) ∣ = 0 = M ∥ z ∥^{2}$ , while if $z \neq 0$ then [A2] gives $q (z) = ∥ z ∥^{2} q (u)$ for the unit vector $u = z / ∥ z ∥$ , hence $∣ q (z) ∣ = ∥ z ∥^{2} ∣ q (u) ∣ \leq M ∥ z ∥^{2}$ .

A2algebra

1.2

$M \leq ∥ T ∥$ . For every unit vector $x$ , Cauchy–Schwarz and the norm bound give $∣ q (x) ∣ = ∣ ⟨ T x, x ⟩ ∣ \leq ∥ T x ∥ ∥ x ∥ \leq ∥ T ∥$ , so $∥ T ∥$ is an upper bound of the set whose supremum is $M$ ; hence $M \leq ∥ T ∥$ , and when $H = {0}$ both numbers are $0$ .

A4algebra

2.1

A sesquilinear bound. For all $x, y \in H$ one has $∣ ⟨ T x, y ⟩ ∣ \leq \frac{M}{2} (∥ x ∥^{2} + ∥ y ∥^{2})$ : if $x = 0$ or $y = 0$ then $⟨ T x, y ⟩ = 0$ and the right side is $\geq 0$ ; otherwise put $c : = ⟨ T x, y ⟩$ , and if $c \neq 0$ choose the unit scalar $ω$ with $\overline{ω} c = ∣ c ∣$ (namely $ω = c / ∣ c ∣$ over $C$ , $ω = sign (c)$ over $R$ ) and set $u : = ω y$ , so that $∥ u ∥ = ∥ y ∥$ and $⟨ T x, u ⟩ = \overline{ω} c = ∣ c ∣ \geq 0$ is real; then [A2] gives $4 ∣ ⟨ T x, y ⟩ ∣ = 4 ⟨ T x, u ⟩ = q (x + u) - q (x - u) \leq ∣ q (x + u) ∣ + ∣ q (x - u) ∣ \leq M (∥ x + u ∥^{2} + ∥ x - u ∥^{2})$ by [step 1.1], and the parallelogram law [A3] turns the last factor into $2 ∥ x ∥^{2} + 2 ∥ u ∥^{2} = 2 ∥ x ∥^{2} + 2 ∥ y ∥^{2}$ .

step 1.1A1A2A3algebra

3.1

Removing the norms. For $x, y \neq 0$ and every real $t > 0$ , [step 2.1] applied to the pair $(t x, t^{- 1} y)$ together with the scaling identity of [A2] gives $∣ ⟨ T x, y ⟩ ∣ \leq \frac{M}{2} (t^{2} ∥ x ∥^{2} + t^{- 2} ∥ y ∥^{2})$ ; the right side is minimised at $t^{2} = ∥ y ∥ / ∥ x ∥ > 0$ , where it equals $M ∥ x ∥ ∥ y ∥$ , so $∣ ⟨ T x, y ⟩ ∣ \leq M ∥ x ∥ ∥ y ∥$ for all $x, y \in H$ (the zero cases being trivial).

step 2.1A2algebra

4.1

$∥ T ∥ \leq M$ . For $x \neq 0$ the dual norm formula [A4] gives $∥ T x ∥ = \sup_{∥ y ∥ = 1} ∣ ⟨ T x, y ⟩ ∣ \leq M ∥ x ∥$ by [step 3.1] with $∥ y ∥ = 1$ , and the inequality also holds at $x = 0$ ; thus $M$ is a uniform bound for $T$ on the unit ball, so $∥ T ∥ \leq M$ by the unit-ball characterisation of the operator norm in [A4].

step 3.1A4algebra

5.1

Conclusion. Steps 1.2 and 4.1 give $∥ T ∥ = M$ ; if $H = {0}$ then $T = 0$ , $M = 0$ by the empty-supremum convention and $∥ T ∥ = 0$ , so the identity holds there as well.

step 1.2step 4.1A5∎

Depends on

Used by

Norm point of a compact self adjoint operator is an eigenvalue up to sign Lemma

Dependency tree · two levels

34 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

Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §1.3, Problems 1.19–1.20 (quadratic-form bound and polarization) (standard reference, not scraped)
Anthony W. Knapp, Advanced Real Analysis — Chapter II, §2, Proposition 2.2 (standard reference, not scraped)