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In [[mathematics]], in the field of [[functional analysis]], the '''Cotlar–Stein almost orthogonality lemma''' is named after mathematicians [[Mischa Cotlar]]
and [[Elias Stein]]. 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<ref>{{harvnb|Cotlar|1955}}</ref> and allowed him to conclude that the [[Hilbert transform]]
is a [[continuous linear operator]] in <math>L^2</math>
without using the [[Fourier transform]].
A more general version was proved by Elias Stein.<ref>{{harvnb|Stein|1993}}</ref>

==Cotlar–Stein almost orthogonality lemma==
Let <math>E,\,F</math> be two [[Hilbert spaces]].
Consider a family of operators
<math>T_j</math>, <math>j\ge 1</math>,
with each <math>T_j</math>
a [[bounded linear operator]] from <math>E</math> to <math>F</math>.

Denote

: <math>a_{jk}=\Vert T_j T_k^\ast\Vert,
\qquad b_{jk}=\Vert T_j^\ast T_k\Vert.</math>

The family of operators
<math>T_j:\;E\to F</math>, <math>j\ge 1,</math>
is ''almost orthogonal'' if

:<math>A=\sup_{j}\sum_{k}\sqrt{a_{jk}}<\infty,
\qquad B=\sup_{j}\sum_{k}\sqrt{b_{jk}}<\infty.</math>

The Cotlar–Stein lemma states that if <math>T_j</math>
are almost orthogonal,
then the series
<math>\sum_{j}T_j</math>
converges in the [[strong operator topology]],
and that

:<math>\Vert \sum_{j}T_j\Vert \le\sqrt{AB}.</math>

==Proof==
If ''R''<sub>1</sub>, ..., ''R''<sub>''n''</sub> is a finite collection of bounded operators, then<ref>{{harvnb|Hörmander|1994}}</ref>

:<math>\displaystyle{\sum_{i,j} |(R_i v,R_jv)| \le \left(\max_i \sum_j \|R_i^*R_j\|^{1\over 2}\right)\left(\max_i \sum_j \|R_iR_j^*\|^{1\over 2}\right)\|v\|^2.}</math>

So under the hypotheses of the lemma,

:<math>\displaystyle{\sum_{i,j} |(T_i v,T_jv)| \le AB\|v\|^2.}</math>

It follows that 

:<math>\displaystyle{\|\sum_{i=1}^n T_iv\|^2 \le AB \|v\|^2,}</math>

and that

:<math>\displaystyle{\|\sum_{i=m}^n T_iv\|^2 \le \sum_{i,j\ge m} |(T_iv,T_jv)|.}</math>

Hence the partial sums 

:<math>\displaystyle{s_n=\sum_{i=1}^n T_iv}</math>

form a [[Cauchy sequence]]. 

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

To prove the inequality above set

:<math>\displaystyle{R=\sum a_{ij}R_i^*R_j}</math>

with |''a''<sub>''ij''</sub>| ≤ 1 chosen so that

:<math>\displaystyle{(Rv,v)=|(Rv,v)|=\sum |(R_iv,R_jv)|.}</math>

Then

:<math>\displaystyle{\|R\|^{2m} =\|(R^*R)^m\|\le \sum \|R_{i_1}^* R_{i_2} R_{i_3}^* R_{i_4} \cdots R_{i_{2m}}\| \le \sum \left(\|R_{i_1}^*\|\|R_{i_1}^*R_{i_2}\|\|R_{i_2}R_{i_3}^*\|\cdots \|R_{i_{2m-1}}^* R_{i_{2m}}\|\|R_{i_{2m}}\|\right)^{1\over 2}.}</math>

Hence

:<math>\displaystyle{\|R\|^{2m} \le n \cdot \max \|R_i\| \left(\max_i \sum_j \|R_i^*R_j\|^{1\over 2}\right)^{2m}\left(\max_i \sum_j \|R_iR_j^*\|^{1\over 2}\right)^{2m}.}</math>

Taking 2''m''th roots and letting ''m'' tend to ≈,

:<math>\displaystyle{\|R\|\le  \left(\max_i \sum_j \|R_i^*R_j\|^{1\over 2}\right)\left(\max_i \sum_j \|R_iR_j^*\|^{1\over 2}\right),}</math>

which immediately implies the inequality.

==Generalization==
There is a generalization of the Cotlar–Stein lemma with sums replaced by integrals.<ref>{{harvnb|Knapp|Stein|1971}}</ref> 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 

:<math>\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),}</math>

are finite, then the function ''T''(''x'')''v'' is integrable for each ''v'' in ''E'' with

:<math>\displaystyle{\|\int_X T(x)v\, d\mu(x)\| \le \sqrt{AB}  \cdot \|v\|.}</math>

The result can be proved by replacing sums by integrals in the previous proof or by using Riemann sums to approximate the integrals.

==Example==
Here is an example of an ''orthogonal'' family of operators.  Consider the inifite-dimensional matrices

:<math>
T=\left[
\begin{array}{cccc}
1&0&0&\vdots\\0&1&0&\vdots\\0&0&1&\vdots\\\cdots&\cdots&\cdots&\ddots\end{array}
\right]
</math>

and also

:<math>
\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.
</math>

Then
<math>\Vert T_j\Vert=1</math> for each <math>j</math>,
hence the series <math>\sum_{j\in\mathbb{N}}T_j</math>
does not converge in the [[uniform operator topology]].

Yet, since
<math>\Vert T_j T_k^\ast\Vert=0</math>
and 
<math>\Vert T_j^\ast T_k\Vert=0</math>
for <math>j\ne k</math>,
the Cotlar–Stein almost orthogonality lemma tells us that 

:<math>T=\sum_{j\in\mathbb{N}}T_j</math>

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

==Notes==
{{reflist|2}}

==References==
*{{citation|first=Mischa|last= Cotlar|title=A combinatorial inequality and its application to L<sup>2</sup> spaces|journal= Math. Cuyana|volume= 1|year=1955|pages= 41–55}}
*{{citation|first=Lars|last=Hörmander|title=Analysis of Partial Differential Operators III: Pseudodifferential Operators|publisher=Springer-Verlag|year=1994|edition=2nd|isbn=978-3-540-49937-4|pages=165–166}}
*{{citation|first=Anthony W.|last=Knapp|first2=Elias|last2=Stein|title=Intertwining operators for semisimple Lie groups|journal=Ann. Math.|volume=93|year=1971|pages=489–579}}
*{{citation|first=Elias|last= Stein|title=Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals|publisher= Princeton University Press|year= 1993|isbn= 0-691-03216-5}}

{{DEFAULTSORT:Cotlar-Stein lemma}}
[[Category:Hilbert space]]
[[Category:Harmonic analysis]]
[[Category:Operator theory]]
[[Category:Inequalities]]
[[Category:Theorems in functional analysis]]