Cotlar–Stein lemma

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^[1] and allowed him to conclude that the Hilbert transform is a continuous linear operator in ${\displaystyle L^{2}}$ without using the Fourier transform. A more general version was proved by Elias Stein.^[2]

Statement of the lemma

Let ${\displaystyle E,\,F}$ be two Hilbert spaces. Consider a family of operators ${\displaystyle T_{j}}$, ${\displaystyle j\geq 1}$, with each ${\displaystyle T_{j}}$ a bounded linear operator from ${\displaystyle E}$ to ${\displaystyle F}$.

Denote

${\displaystyle a_{jk}=\Vert T_{j}T_{k}^{\ast }\Vert ,\qquad b_{jk}=\Vert T_{j}^{\ast }T_{k}\Vert .}$

The family of operators ${\displaystyle T_{j}:\;E\to F}$, ${\displaystyle j\geq 1,}$ is almost orthogonal if

${\displaystyle A=\sup _{j}\sum _{k}{\sqrt {a_{jk}}}<\infty ,\qquad B=\sup _{j}\sum _{k}{\sqrt {b_{jk}}}<\infty .}$

The Cotlar–Stein lemma states that if ${\displaystyle T_{j}}$ are almost orthogonal, then the series ${\displaystyle \sum _{j}T_{j}}$ converges in the strong operator topology, and

${\displaystyle \Vert \sum _{j}T_{j}\Vert \leq {\sqrt {AB}}.}$

Proof

If ${\displaystyle T_{1},\ldots ,T_{n}}$ is a finite collection of bounded operators, then^[3] (as will be proven below)

${\displaystyle \displaystyle {\sum _{i,j}|(T_{i}v,T_{j}v)|\leq \left(\max _{i}\sum _{j}\|T_{i}^{*}T_{j}\|^{1 \over 2}\right)\left(\max _{i}\sum _{j}\|T_{i}T_{j}^{*}\|^{1 \over 2}\right)\|v\|^{2}.}}$

So under the hypotheses of the lemma,

${\displaystyle \displaystyle {\sum _{i,j}|(T_{i}v,T_{j}v)|\leq AB\|v\|^{2}.}}$

It follows that

${\displaystyle \displaystyle {\|\sum _{i=1}^{n}T_{i}v\|^{2}\leq AB\|v\|^{2},}}$

and that

${\displaystyle \displaystyle {\|\sum _{i=m}^{n}T_{i}v\|^{2}\leq \sum _{i,j\geq m}|(T_{i}v,T_{j}v)|.}}$

Hence, the partial sums

${\displaystyle \displaystyle {s_{n}=\sum _{i=1}^{n}T_{i}v}}$

form a Cauchy sequence.

The sum is therefore absolutely convergent with the limit satisfying the stated inequality.

To prove the inequality above set

${\displaystyle \displaystyle {R=\sum a_{ij}T_{i}^{*}T_{j}}}$

with |a_ij| ≤ 1 chosen so that

${\displaystyle \displaystyle {(Rv,v)=|(Rv,v)|=\sum |(T_{i}v,T_{j}v)|.}}$

Then

${\displaystyle \displaystyle {\|R\|^{2m}=\|(R^{*}R)^{m}\|\leq \sum \|T_{i_{1}}^{*}T_{i_{2}}T_{i_{3}}^{*}T_{i_{4}}\cdots T_{i_{4m}}\|\leq \sum \left(\|T_{i_{1}}^{*}\|\|T_{i_{1}}^{*}T_{i_{2}}\|\|T_{i_{2}}T_{i_{3}}^{*}\|\cdots \|T_{i_{4m-1}}^{*}T_{i_{4m}}\|\|T_{i_{4m}}\|\right)^{1 \over 2}.}}$

Hence

${\displaystyle \displaystyle {\|R\|^{2m}\leq n\cdot \max \|T_{i}\|\left(\max _{i}\sum _{j}\|T_{i}^{*}T_{j}\|^{1 \over 2}\right)^{2m}\left(\max _{i}\sum _{j}\|T_{i}T_{j}^{*}\|^{1 \over 2}\right)^{2m-1}.}}$

Taking 2mth roots and letting m tend to ∞,

${\displaystyle \displaystyle {\|R\|\leq \left(\max _{i}\sum _{j}\|T_{i}^{*}T_{j}\|^{1 \over 2}\right)\left(\max _{i}\sum _{j}\|T_{i}T_{j}^{*}\|^{1 \over 2}\right),}}$

which immediately implies the inequality.

Generalization

The Cotlar-Stein lemma has been generalized, with sums being replaced by integrals.^[4][5] 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

${\displaystyle \displaystyle {A=\sup _{x}\int _{X}\|T(x)^{*}T(y)\|^{1 \over 2}\,d\mu (y),\,\,\,B=\sup _{x}\int _{X}\|T(y)T(x)^{*}\|^{1 \over 2}\,d\mu (y),}}$

are finite, then the function T(x)v is integrable for each v in E with

${\displaystyle \displaystyle {\|\int _{X}T(x)v\,d\mu (x)\|\leq {\sqrt {AB}}\cdot \|v\|.}}$

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.

${\displaystyle T=\left[{\begin{array}{cccc}1&0&0&\vdots \\0&1&0&\vdots \\0&0&1&\vdots \\\cdots &\cdots &\cdots &\ddots \end{array}}\right]}$

and also

${\displaystyle \qquad T_{1}=\left[{\begin{array}{cccc}1&0&0&\vdots \\0&0&0&\vdots \\0&0&0&\vdots \\\cdots &\cdots &\cdots &\ddots \end{array}}\right],\qquad T_{2}=\left[{\begin{array}{cccc}0&0&0&\vdots \\0&1&0&\vdots \\0&0&0&\vdots \\\cdots &\cdots &\cdots &\ddots \end{array}}\right],\qquad T_{3}=\left[{\begin{array}{cccc}0&0&0&\vdots \\0&0&0&\vdots \\0&0&1&\vdots \\\cdots &\cdots &\cdots &\ddots \end{array}}\right],\qquad \dots .}$

Then ${\displaystyle \Vert T_{j}\Vert =1}$ for each ${\displaystyle j}$, hence the series ${\displaystyle \sum _{j\in \mathbb {N} }T_{j}}$ does not converge in the uniform operator topology.

Yet, since ${\displaystyle \Vert T_{j}T_{k}^{\ast }\Vert =0}$ and ${\displaystyle \Vert T_{j}^{\ast }T_{k}\Vert =0}$ for ${\displaystyle j\neq k}$, the Cotlar–Stein almost orthogonality lemma tells us that

${\displaystyle T=\sum _{j\in \mathbb {N} }T_{j}}$

converges in the strong operator topology and is bounded by 1.