Coburn's alternative for Toeplitz operators

For a nonzero essentially bounded symbol a on the unit circle, either the Toeplitz operator T_a or its adjoint T_a† = T_ā is injective. Consequently a Fredholm Toeplitz operator of index zero is invertible; this is how invertibility of T_a for a continuous symbol a reduces to the computation of its Fredholm index.

The proof is the Hardy product argument. A Hardy function f lies in the kernel of T_a exactly when a f has only negative Fourier modes (TauCeti.toeplitz_apply_eq_zero_iff). Suppose that T_a f = 0 and T_ā g = 0. The conjugate conj g of a Hardy function has only nonpositive modes, so the integrable product a f conj g has only negative modes. Its complex conjugate ā g conj f has only negative modes for the same reason, so a f conj g also has only positive modes. Hence all Fourier coefficients of a f conj g vanish and a f conj g = 0 almost everywhere. If f and g are nonzero, then by boundary uniqueness they are nonzero almost everywhere, which forces a = 0.

Main results

• TauCeti.eq_zero_of_toeplitz_apply_eq_zero_of_toeplitz_star_apply_eq_zero: if T_a and T_ā both have nonzero kernel vectors, then a = 0.
• TauCeti.ker_toeplitz_eq_bot_or_ker_adjoint_toeplitz_eq_bot: Coburn's alternative: for a ≠ 0, ker T_a = 0 or ker T_a† = 0.
• TauCeti.isInvertible_toeplitz_of_isFredholm_of_index_eq_zero: a Fredholm Toeplitz operator of index zero is invertible.

References

• L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288.
• E. Park, Toeplitz operators, for the Hardy product argument.
• A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, second edition, Springer (2006), Chapter 2.

theorem TauCeti.eq_zero_of_toeplitz_apply_eq_zero_of_toeplitz_star_apply_eq_zero {a : ↥(MeasureTheory.Lp ℂ ⊤ AddCircle.haarAddCircle)} {f g : HardySpace} (hf₀ : f ≠ 0) (hg₀ : g ≠ 0) (hf : (toeplitz a) f = 0) (hg : (toeplitz (star a)) g = 0) : a = 0

The Hardy product argument. If the Toeplitz operators T_a and T_ā = T_a† have nonzero kernel vectors f and g, then the symbol a is zero.

theorem TauCeti.ker_toeplitz_eq_bot_or_ker_adjoint_toeplitz_eq_bot {a : ↥(MeasureTheory.Lp ℂ ⊤ AddCircle.haarAddCircle)} (ha : a ≠ 0) : (↑(toeplitz a)).ker = ⊥ ∨ (↑(ContinuousLinearMap.adjoint (toeplitz a))).ker = ⊥

Coburn's alternative. For a nonzero essentially bounded symbol a, either the Toeplitz operator T_a or its adjoint has trivial kernel.

theorem TauCeti.isInvertible_toeplitz_of_isFredholm_of_index_eq_zero {a : ↥(MeasureTheory.Lp ℂ ⊤ AddCircle.haarAddCircle)} (hT : (toeplitz a).IsFredholm) (h₀ : (toeplitz a).index = 0) : (toeplitz a).IsInvertible

A Fredholm Toeplitz operator of index zero is invertible.