The Brown–Halmos characterization of Toeplitz operators

The forward shift on boundary Hardy space is the Toeplitz operator S = T_z of the Fourier monomial z = e₁; it sends the Hardy basis vector eₙ to eₙ₊₁. A bounded operator T on H² is a Toeplitz operator T_a for an essentially bounded symbol a exactly when its matrix in the Hardy basis is constant along diagonals, that is, exactly when S† T S = T; the symbol is then unique. This is the theorem of Brown and Halmos, and it is the way an abstractly given operator is recognized as a Toeplitz operator.

• TauCeti.exists_toeplitz_eq_iff_inner_hardyBasis_succ: T is a Toeplitz operator if and only if ⟪e_{i+1}, T e_{j+1}⟫ = ⟪e_i, T e_j⟫ for all i, j.
• TauCeti.existsUnique_toeplitz_eq_iff_adjoint_comp_comp_eq: Brown–Halmos. T = T_a for a unique essentially bounded symbol a if and only if S† T S = T.

For the converse direction, write c_{i-j} = ⟪e_i, T e_j⟫. The circle function b = T e₀ + conj (T† e₀) - c₀, built from the first column and the first row of the matrix, has Fourier coefficients b̂(k) = c_k for every k ∈ ℤ. For trigonometric polynomials f and q, multiplying both by a sufficiently high Fourier mode e_N moves them into H², and then ⟪f, b q⟫ = ⟪e_N f, T (e_N q)⟫, so ‖b q‖₂ ≤ ‖T‖ ‖q‖₂. The multiplier criterion TauCeti.eLpNorm_top_le_of_forall_mem_span_fourier therefore makes b essentially bounded, and T_b has the same matrix as T.

References

• A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1964), 89–102, Theorem 6.
• A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, second edition, §2.

theorem TauCeti.exists_toeplitz_eq_iff_inner_hardyBasis_succ (T : HardySpace →L[ℂ] HardySpace) : (∃ (a : ↥(MeasureTheory.Lp ℂ ⊤ AddCircle.haarAddCircle)), toeplitz a = T) ↔ ∀ (i j : ℕ), inner ℂ (hardyBasis (i + 1)) (T (hardyBasis (j + 1))) = inner ℂ (hardyBasis i) (T (hardyBasis j))

An operator on Hardy space is a Toeplitz operator T_a for an essentially bounded symbol a if and only if its matrix in the Hardy basis is constant along diagonals.

theorem TauCeti.existsUnique_toeplitz_eq_iff_adjoint_comp_comp_eq (T : HardySpace →L[ℂ] HardySpace) : (∃! a : ↥(MeasureTheory.Lp ℂ ⊤ AddCircle.haarAddCircle), toeplitz a = T) ↔ ContinuousLinearMap.adjoint (toeplitz (fourierLp ⊤ 1)) ∘SL T ∘SL toeplitz (fourierLp ⊤ 1) = T

Brown–Halmos theorem. A bounded operator T on Hardy space is a Toeplitz operator T_a for a unique essentially bounded symbol a if and only if S† T S = T, where S = T_z is the forward shift.