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The Hilbert transform is skew-adjoint on L2

Statement

Assume Countable Choice and use the first-variable-linear pairing $⟨ f, g ⟩ = \int_{R} f \overline{g} d x$ on $L^{2} (R; C)$ . Then for all $f, g \in L^{2} (R; C)$ ,

$⟨ H f, g ⟩ = - ⟨ f, H g ⟩ .$

Equivalently $H^{*} = - H$ : the Hilbert transform is skew-adjoint, and the statement is a statement about the $L^{2}$ extension of the Schwartz principal-value operator, not about pointwise values.

Facts & Assumptions

Given: Countable Choice, the first-variable-linear pairing, and the $L^{2}$ Hilbert transform $H$ with symbol $m (ξ) = - i sgn (ξ)$ .

[F1]

The Hilbert transform is the $L^{2}$ operator with multiplier $m (ξ) = - i sgn (ξ)$ , extending the Schwartz principal-value operator; $∥ H f ∥_{2} = ∥ f ∥_{2}$ and $H^{2} = - I$ . The Hilbert transform is an L2 isometry and squares to minus the identity

[F2]

For a measurable symbol with essential supremum at most one the operator $F_{2}^{- 1} M_{m} F_{2}$ acts on $L^{2}$ , and the Schwartz-core action of $H$ extends uniquely to it. Exact L2 Fourier multiplier norm

[F3]

Plancherel: $F_{2}$ is a surjective linear isometry that preserves the first-variable-linear inner product, $⟨ f, g ⟩ = ⟨ F_{2} f, F_{2} g ⟩$ . Plancherel theorem

Proof

technique · direct

1.1givenalgebra

The symbol satisfies $\overline{m (ξ)} = - m (ξ)$ for every $ξ \neq 0$ : indeed $\overline{- i sgn ξ} = i sgn ξ = - (- i sgn ξ)$ , and both sides vanish at $ξ = 0$ .

2.1step 1.1F1F2F3

Since $F_{2}$ preserves the inner product by [F3] and $H$ is the multiplier operator of [F1] with $F_{2} (H f) = m F_{2} f$ , one has $⟨ H f, g ⟩ = ⟨ m F_{2} f, F_{2} g ⟩ = \int m \hat{f} \overline{\hat{g}} d λ$ for all $f, g \in L^{2}$ , the last expression being an absolutely convergent integral because $∣ m ∣ \leq 1$ and $\hat{f}, \hat{g} \in L^{2}$ .

3.1step 1.1step 2.1F3∎

Applying 2.1 with the roles of $f$ and $g$ interchanged and conjugating the symbol by 1.1, $⟨ f, H g ⟩ = \int \hat{f} \overline{m \hat{g}} d λ = \int \hat{f} \overline{m} \overline{\hat{g}} d λ = - \int m \hat{f} \overline{\hat{g}} d λ = - ⟨ H f, g ⟩$ , which is the asserted skew-adjointness.

Depends on

Used by

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Sources

Loukas Grafakos, Classical Fourier Analysis, third edition (standard reference, not scraped)
Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)