The Cotlar–Stein almost-orthogonality lemma

Let T i : E →L[𝕜] F be bounded operators between Hilbert spaces. If ∑ j, √‖(T i)† ∘L T j‖ ≤ A and ∑ j, √‖T i ∘L (T j)†‖ ≤ A for every i, then the operator norm of ∑ i, T i is at most A, although the triangle inequality only gives the bound ∑ i, ‖T i‖, which may be as large as A times the number of summands. For infinitely many operators the series ∑ i, T i x then converges for every x, and its sum is a bounded operator of norm at most A.

This is the almost-orthogonality principle behind the L² boundedness of pseudodifferential operators with symbols of order zero (Calderón–Vaillancourt): such an operator is decomposed dyadically, and the pieces are almost orthogonal in the above sense.

Main results

• TauCeti.norm_sum_le_of_almost_orthogonal: the finite Cotlar–Stein lemma.
• TauCeti.exists_hasSum_apply_of_almost_orthogonal: the Cotlar–Stein lemma for an arbitrary family; the series converges in the strong operator topology to an operator of norm at most A.

Implementation notes

For a finite family with sum S, the C⋆-identity gives ‖S‖ ^ (2 * 2 ^ k) = ‖(S† S) ^ 2 ^ k‖. Expanding (S† S) ^ n = (∑ i j, (T i)† T j) ^ n gives a sum of words (T i₁)† T j₁ ⋯ (T iₙ)† T jₙ. The norm of each word is bounded both by the product of the norms of its factors (T iₘ)† T jₘ and by ‖T i₁‖ ‖T j₁ (T i₂)†‖ ⋯ ‖T jₙ‖, hence by the geometric mean of these two bounds. Summing the geometric means one index at a time bounds ‖(S† S) ^ n‖ by card ι * A ^ (2 * n), and letting n → ∞ removes the factor card ι.

For an infinite family, the vectors on which the series converges form a closed subspace, by the uniform bound on the finite partial sums (ContinuousLinearMap.isClosed_setOf_summable_apply). It contains the range of every (T j)†, on which the series converges absolutely, so its orthogonal complement is killed by every T j, hence is zero.

References

• M. Cotlar, A combinatorial inequality and its applications to L²-spaces, Rev. Mat. Cuyana 1 (1955), 41–55.
• A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. 93 (1971), 489–578.
• E. M. Stein, Harmonic Analysis, Princeton University Press, 1993, Chapter VII, §2.

theorem TauCeti.norm_sum_le_of_almost_orthogonal {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {ι : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (T : ι → E →L[𝕜] F) {A : ℝ} (s : Finset ι) (hA : 0 ≤ A) (h₁ : ∀ i ∈ s, ∑ j ∈ s, √‖ContinuousLinearMap.adjoint (T i) ∘SL T j‖ ≤ A) (h₂ : ∀ i ∈ s, ∑ j ∈ s, √‖T i ∘SL ContinuousLinearMap.adjoint (T j)‖ ≤ A) : ‖∑ i ∈ s, T i‖ ≤ A

The Cotlar–Stein lemma, finite form. If the operators T i, i ∈ s, between Hilbert spaces are almost orthogonal in the sense that ∑ j ∈ s, √‖(T i)† ∘L T j‖ ≤ A and ∑ j ∈ s, √‖T i ∘L (T j)†‖ ≤ A for every i ∈ s, then ‖∑ i ∈ s, T i‖ ≤ A.

theorem TauCeti.exists_hasSum_apply_of_almost_orthogonal {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {ι : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (T : ι → E →L[𝕜] F) {A : ℝ} (hA : 0 ≤ A) (h₁ : ∀ (i : ι) (s : Finset ι), ∑ j ∈ s, √‖ContinuousLinearMap.adjoint (T i) ∘SL T j‖ ≤ A) (h₂ : ∀ (i : ι) (s : Finset ι), ∑ j ∈ s, √‖T i ∘SL ContinuousLinearMap.adjoint (T j)‖ ≤ A) : ∃ (S : E →L[𝕜] F), ‖S‖ ≤ A ∧ ∀ (x : E), HasSum (fun (i : ι) => (T i) x) (S x)

The Cotlar–Stein lemma. Let T i be operators between Hilbert spaces which are almost orthogonal in the sense that ∑ j, √‖(T i)† ∘L T j‖ ≤ A and ∑ j, √‖T i ∘L (T j)†‖ ≤ A for every i, the sums over j being taken over arbitrary finite sets. Then for every x the series ∑ i, T i x converges unconditionally, and its sum is a bounded operator of norm at most A.