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A self-adjoint operator is detected by its quadratic form

Statement

Let $K$ be a complex Hilbert space and let $S \in B (K)$ be self-adjoint (Self-adjoint, positive, unitary and normal operators, Hilbert space, A bounded linear operator between normed spaces). Then $∥ S ∥ = \sup_{∥ η ∥ = 1} ∣ ⟨ S η, η ⟩ ∣,$ and if $S \geq 0$ (that is, $⟨ S η, η ⟩ \geq 0$ for every $η$ ) then $∥ S ∥ = \sup_{∥ η ∥ = 1} ⟨ S η, η ⟩ .$ The supremum is taken over the unit sphere of $K$ ; when $K = {0}$ the supremum over the empty set is understood as $0$ in $[0, \infty)$ , and the statements read $0 = 0$ . No attainment of the supremum is asserted.

Facts & Assumptions

Given: a complex Hilbert space $K$ and a self-adjoint bounded operator $S \in B (K)$ .

[A1]

The pairing is linear in the first argument and conjugate-linear in the second, and $⟨ v, v ⟩ = ∥ v ∥^{2}$ (Hilbert space, Real and complex inner-product spaces and their induced length). The operator norm satisfies $∥ T v ∥ \leq ∥ T ∥ ∥ v ∥$ and, when $K \neq {0}$ , $∥ T ∥ = \sup_{∥ v ∥ = 1} ∥ T v ∥$ ; when $K = {0}$ , $∥ T ∥ = 0$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[A2]

Cauchy–Schwarz: $∣ ⟨ v, w ⟩ ∣ \leq ∥ v ∥ ∥ w ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[A3]

The parallelogram law holds: $∥ v + w ∥^{2} + ∥ v - w ∥^{2} = 2 ∥ v ∥^{2} + 2 ∥ w ∥^{2}$ (The parallelogram law).

[A4]

$S$ is self-adjoint, so $⟨ S v, w ⟩ = ⟨ v, S w ⟩$ for all $v, w$ by the defining identity of its adjoint (The Hilbert-space adjoint of a bounded operator); and $S \geq 0$ means $⟨ S η, η ⟩ \geq 0$ for every $η$ (Self-adjoint, positive, unitary and normal operators). Only the given self-adjoint operator and its defining identity are used; existence of adjoints for arbitrary operators is not invoked.

Proof

technique · direct

Given: a complex Hilbert space $K$ , a self-adjoint $S \in B (K)$ , and the number $M : = \sup_{∥ η ∥ = 1} ∣ ⟨ S η, η ⟩ ∣$ with value $0$ when $K = {0}$ .

1.1A1A2

$M \leq ∥ S ∥$ : for unit $η$ , Cauchy–Schwarz and $∥ S η ∥ \leq ∥ S ∥$ give $∣ ⟨ S η, η ⟩ ∣ \leq ∥ S η ∥ \leq ∥ S ∥$ .

1.2A1A4

For unit $x, y$ , self-adjointness gives $⟨ S (x + y), x + y ⟩ = ⟨ S x, x ⟩ + ⟨ S x, y ⟩ + ⟨ S y, x ⟩ + ⟨ S y, y ⟩$ and $⟨ S (x - y), x - y ⟩ = ⟨ S x, x ⟩ - ⟨ S x, y ⟩ - ⟨ S y, x ⟩ + ⟨ S y, y ⟩$ , and $⟨ S y, x ⟩ = \overline{⟨ S x, y ⟩}$ ; subtracting, $4 Re ⟨ S x, y ⟩ = ⟨ S (x + y), x + y ⟩ - ⟨ S (x - y), x - y ⟩$ .

2.1A3step 1.2

For unit $x, y$ one has $∣ ⟨ S x, y ⟩ ∣ \leq M$ : if $u : = ⟨ S x, y ⟩ \neq 0$ , replace $y$ by the unit vector $y^{'} = (u / ∣ u ∣) y$ , so that $⟨ S x, y^{'} ⟩ = (\overline{u} / ∣ u ∣) ⟨ S x, y ⟩ = ∣ u ∣$ is real and nonnegative; then step 1.2 applies to $(x, y^{'})$ , and bounding each quadratic form by $M$ times the squared norm by rescaling nonzero vectors (the quadratic form at zero is zero) and applying the parallelogram law gives $4 ∣ u ∣ = 4 Re ⟨ S x, y^{'} ⟩ \leq M (∥ x + y^{'} ∥^{2} + ∥ x - y^{'} ∥^{2}) = 4 M$ .

3.1A1step 2.1

For unit $x$ one has $∥ S x ∥ \leq M$ : if $S x = 0$ this is clear, and otherwise $y : = S x / ∥ S x ∥$ is a unit vector with $∣ ⟨ S x, y ⟩ ∣ = ∥ S x ∥$ , so step 2.1 applies; consequently $∥ S ∥ = \sup_{∥ x ∥ = 1} ∥ S x ∥ \leq M$ by [A1].

4.1A1A4step 1.1step 3.1∎

Steps 1.1 and 3.1 give $M = ∥ S ∥$ , which is the first display. If $S \geq 0$ , then $⟨ S η, η ⟩ \geq 0$ for every $η$ by [A4], so $∣ ⟨ S η, η ⟩ ∣ = ⟨ S η, η ⟩$ for every $η$ and the same supremum equals $\sup_{∥ η ∥ = 1} ⟨ S η, η ⟩$ , giving the second display. When $K = {0}$ both suprema are the empty supremum $0$ by the stated convention and $∥ S ∥ = 0$ .

Depends on

Used by

Dependency tree · two levels

28 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

Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (Cambridge University Press 2008; author-hosted complete text) (standard reference, not scraped)
Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019) (standard reference, not scraped)