The Cotlar–Stein almost orthogonality lemma is a mathematical lemma in the field of functional analysis. It may be used to obtain information on the operator norm on an operator, acting from one Hilbert space into another, when the operator can be decomposed into almost orthogonal pieces. The original version of this lemma (for self-adjoint and mutually commuting operators) was proved by Mischa Cotlar in 1955 and allowed him to conclude that the Hilbert transform is a continuous linear operator in L^2 without using the Fourier transform. A more general version was proved by Elias Stein.

Statement of the lemma

Let E,,F be two Hilbert spaces. Consider a family of operators T_j, j\geq 1, with each T_j a bounded linear operator from E to F. Denote The family of operators, j\ge 1, is almost orthogonal if The Cotlar–Stein lemma states that if T_j are almost orthogonal, then the series \sum_{j}T_j converges in the strong operator topology, and

Proof

If is a finite collection of bounded operators, then So under the hypotheses of the lemma, It follows that and that Hence, the partial sums form a Cauchy sequence. The sum is therefore absolutely convergent with the limit satisfying the stated inequality. To prove the inequality above set with |aij| ≤ 1 chosen so that Then Hence Taking 2mth roots and letting m tend to ∞, which immediately implies the inequality.

Generalization

The Cotlar-Stein lemma has been generalized, with sums being replaced by integrals. Let X be a locally compact space and μ a Borel measure on X. Let T(x) be a map from X into bounded operators from E to F which is uniformly bounded and continuous in the strong operator topology. If are finite, then the function T(x)v is integrable for each v in E with The result can be proven by replacing sums with integrals in the previous proof, or by utilizing Riemann sums to approximate the integrals.

Example

Here is an example of an orthogonal family of operators. Consider the infinite-dimensional matrices. and also Then for each j, hence the series does not converge in the uniform operator topology. Yet, since and for j\ne k, the Cotlar–Stein almost orthogonality lemma tells us that converges in the strong operator topology and is bounded by 1.