Laurent operators on circle L²

A Laurent operator is multiplication by an essentially bounded symbol on L² of the circle; in the Fourier basis e_n = exp (2πinx/T) it has the bi-infinite Toeplitz matrix of the Fourier coefficients of its symbol. Concretely, ⟪e_m, e_n b⟫ = b̂(m - n) for every b ∈ L² (TauCeti.inner_fourierBasis_multiplicationOperator_fourierLp). This file characterizes Laurent operators in two ways.

• TauCeti.eLpNorm_top_le_of_forall_mem_span_fourier: the multiplier criterion. If a measurable function a satisfies ‖a q‖_p ≤ C ‖q‖_p for every trigonometric polynomial q, for some 1 ≤ p < ∞, then a is essentially bounded with ‖a‖∞ ≤ C. No integrability of a is assumed.
• TauCeti.commute_multiplicationOperator_fourierLp_one_iff: a bounded operator on circle L² commutes with the bilateral shift, multiplication by e_1, if and only if it is the multiplication operator of an essentially bounded symbol.

For the second statement, an operator W commuting with the bilateral shift sends e_n to e_n b with b = W e_0 (TauCeti.apply_fourierBasis_of_commute), hence sends every trigonometric polynomial q to b q (TauCeti.coeFn_apply_toLp_of_commute). Since ‖b q‖₂ ≤ ‖W‖ ‖q‖₂, the multiplier criterion makes b essentially bounded, and W agrees with multiplication by b on the dense span of the Fourier basis.

These are the characterizations of bounded Laurent matrices through which Toeplitz operators on the Hardy space are recognized, as in the Brown–Halmos theorem. See A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1964), and A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, second edition, §2.

theorem TauCeti.eLpNorm_top_le_of_forall_mem_span_fourier {T : ℝ} [Fact (0 < T)] {p : ENNReal} [Fact (1 ≤ p)] (hp : p ≠ ⊤) {a : AddCircle T → ℂ} (ha : MeasureTheory.AEStronglyMeasurable a AddCircle.haarAddCircle) {C : ENNReal} (h : ∀ q ∈ Submodule.span ℂ (Set.range fourier), MeasureTheory.eLpNorm (a * ⇑q) p AddCircle.haarAddCircle ≤ C * MeasureTheory.eLpNorm (⇑q) p AddCircle.haarAddCircle) : MeasureTheory.eLpNorm a ⊤ AddCircle.haarAddCircle ≤ C

Multiplier criterion on the circle. If multiplication by a measurable function a on the circle satisfies ‖a q‖_p ≤ C ‖q‖_p for every trigonometric polynomial q, for some exponent 1 ≤ p < ∞, then a is essentially bounded by C.

theorem TauCeti.inner_fourierBasis_multiplicationOperator_fourierLp {T : ℝ} [Fact (0 < T)] (b : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)) (m n : ℤ) : inner ℂ (fourierBasis m) ((MeasureTheory.Lp.multiplicationOperator (fourierLp ⊤ n)) b) = fourierCoeff (↑↑b) (m - n)

The matrix coefficient ⟪e_m, e_n b⟫ of multiplication by the Fourier monomial e_n is the Fourier coefficient b̂(m - n).

theorem TauCeti.apply_fourierBasis_of_commute {T : ℝ} [Fact (0 < T)] {W : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)} (hW : Commute W (MeasureTheory.Lp.multiplicationOperator (fourierLp ⊤ 1))) (n : ℤ) : W (fourierBasis n) = fourierLp ⊤ n • W (fourierBasis 0)

An operator on circle L² commuting with the bilateral shift, multiplication by e_1, sends each Fourier mode e_n to e_n b, where b = W e_0.

theorem TauCeti.coeFn_apply_toLp_of_commute {T : ℝ} [Fact (0 < T)] {W : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)} (hW : Commute W (MeasureTheory.Lp.multiplicationOperator (fourierLp ⊤ 1))) {q : C(AddCircle T, ℂ)} (hq : q ∈ Submodule.span ℂ (Set.range fourier)) : ↑↑(W ((ContinuousMap.toLp 2 AddCircle.haarAddCircle ℂ) q)) =ᵐ[AddCircle.haarAddCircle] ↑↑(W (fourierBasis 0)) * ⇑q

An operator on circle L² commuting with the bilateral shift, multiplication by e_1, multiplies every trigonometric polynomial q by b = W e_0.

theorem TauCeti.commute_multiplicationOperator_fourierLp_one_iff {T : ℝ} [Fact (0 < T)] {W : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle) →L[ℂ] ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)} : Commute W (MeasureTheory.Lp.multiplicationOperator (fourierLp ⊤ 1)) ↔ ∃ (a : ↥(MeasureTheory.Lp ℂ ⊤ AddCircle.haarAddCircle)), W = MeasureTheory.Lp.multiplicationOperator a

Commutant of the bilateral shift. A bounded operator on circle L² commutes with the bilateral shift, multiplication by the Fourier monomial e_1, if and only if it is multiplication by an essentially bounded symbol.