Cotlar–Stein lemma

Let $E,\backslash ,F$ be two Hilbert spaces. Consider a family of operators $T\_j$, $j\backslash geq\; 1$, with each $T\_j$ a bounded linear operator from $E$ to $F$.

Denote

: $a\_\{jk\}=\backslash Vert\; T\_j\; T\_k^\backslash ast\backslash Vert,$

\qquad b_{jk}=\Vert T_j^\ast T_k\Vert.

The family of operators $T\_j:\backslash ;E\backslash to\; F$, $j\backslash ge\; 1,$ is almost orthogonal if

:

\qquad B=\sup_{j}\sum_{k}\sqrt{b_{jk}}<\infty.

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

:$\backslash Vert\; \backslash sum\_\{j\}T\_j\backslash Vert\; \backslash le\backslash sqrt\{AB\}.$

If $T\_1,\backslash ldots,T\_n$ is a finite collection of bounded operators, then{{harvnb|Hörmander|1994}}

:$\backslash displaystyle\{\backslash sum\_\{i,j\}\; |(T\_i\; v,T\_jv)|\; \backslash le\; \backslash left(\backslash max\_i\; \backslash sum\_j\; \backslash |T\_i^*T\_j\backslash |^\{1\backslash over\; 2\}\backslash right)\backslash left(\backslash max\_i\; \backslash sum\_j\; \backslash |T\_iT\_j^*\backslash |^\{1\backslash over\; 2\}\backslash right)\backslash |v\backslash |^2.\}$

So under the hypotheses of the lemma,

:$\backslash displaystyle\{\backslash sum\_\{i,j\}\; |(T\_i\; v,T\_jv)|\; \backslash le\; AB\backslash |v\backslash |^2.\}$

It follows that

:$\backslash displaystyle\{\backslash |\backslash sum\_\{i=1\}^n\; T\_iv\backslash |^2\; \backslash le\; AB\; \backslash |v\backslash |^2,\}$

and that

:$\backslash displaystyle\{\backslash |\backslash sum\_\{i=m\}^n\; T\_iv\backslash |^2\; \backslash le\; \backslash sum\_\{i,j\backslash ge\; m\}\; |(T\_iv,T\_jv)|.\}$

Hence, the partial sums

:$\backslash displaystyle\{s\_n=\backslash sum\_\{i=1\}^n\; T\_iv\}$

form a Cauchy sequence.

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

To prove the inequality above set

:$\backslash displaystyle\{R=\backslash sum\; a\_\{ij\}T\_i^*T\_j\}$

with |a_ij| ≤ 1 chosen so that

:$\backslash displaystyle\{(Rv,v)=|(Rv,v)|=\backslash sum\; |(T\_iv,T\_jv)|.\}$

Then

:$\backslash displaystyle\{\backslash |R\backslash |^\{2m\}\; =\backslash |(R^*R)^m\backslash |\backslash le\; \backslash sum\; \backslash |T\_\{i\_1\}^*\; T\_\{i\_2\}\; T\_\{i\_3\}^*\; T\_\{i\_4\}\; \backslash cdots\; T\_\{i\_\{4m\}\}\backslash |\; \backslash le\; \backslash sum\; \backslash left(\backslash |T\_\{i\_1\}^*\backslash |\backslash |T\_\{i\_1\}^*T\_\{i\_2\}\backslash |\backslash |T\_\{i\_2\}T\_\{i\_3\}^*\backslash |\backslash cdots\; \backslash |T\_\{i\_\{4m-1\}\}^*\; T\_\{i\_\{4m\}\}\backslash |\backslash |T\_\{i\_\{4m\}\}\backslash |\backslash right)^\{1\backslash over\; 2\}.\}$

Hence

:$\backslash displaystyle\{\backslash |R\backslash |^\{2m\}\; \backslash le\; n\; \backslash cdot\; \backslash max\; \backslash |T\_i\backslash |\; \backslash left(\backslash max\_i\; \backslash sum\_j\; \backslash |T\_i^*T\_j\backslash |^\{1\backslash over\; 2\}\backslash right)^\{2m\}\backslash left(\backslash max\_i\; \backslash sum\_j\; \backslash |T\_iT\_j^*\backslash |^\{1\backslash over\; 2\}\backslash right)^\{2m-1\}.\}$

Taking 2mth roots and letting m tend to ∞,

:$\backslash displaystyle\{\backslash |R\backslash |\backslash le\; \backslash left(\backslash max\_i\; \backslash sum\_j\; \backslash |T\_i^*T\_j\backslash |^\{1\backslash over\; 2\}\backslash right)\backslash left(\backslash max\_i\; \backslash sum\_j\; \backslash |T\_iT\_j^*\backslash |^\{1\backslash over\; 2\}\backslash right),\}$

which immediately implies the inequality.

The Cotlar-Stein lemma has been generalized, with sums being replaced by integrals.{{harvnb|Knapp|Stein|1971}}{{cite journal|last1=Calderon|first1=Alberto|last2=Vaillancourt|first2=Remi|title=On the boundedness of pseudo-differential operators|journal=Journal of the Mathematical Society of Japan|date=1971|volume=23|issue=2|pages=374–378|doi=10.2969/jmsj/02320374|doi-access=free}} 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

:$\backslash displaystyle\{A=\; \backslash sup\_x\; \backslash int\_X\; \backslash |T(x)^*T(y)\backslash |^\{1\backslash over\; 2\}\; \backslash ,\; d\backslash mu(y),\backslash ,\backslash ,\backslash ,\; B=\; \backslash sup\_x\; \backslash int\_X\; \backslash |T(y)T(x)^*\backslash |^\{1\backslash over\; 2\}\backslash ,\; d\backslash mu(y),\}$

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

:$\backslash displaystyle\{\backslash |\backslash int\_X\; T(x)v\backslash ,\; d\backslash mu(x)\backslash |\; \backslash le\; \backslash sqrt\{AB\}\; \backslash cdot\; \backslash |v\backslash |.\}$

The result can be proven by replacing sums with integrals in the previous proof, or by utilizing Riemann sums to approximate the integrals.

Here is an example of an orthogonal family of operators. Consider the infinite-dimensional matrices.

:$$

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

:$$

\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 $\backslash Vert\; T\_j\backslash Vert=1$ for each $j$, hence the series $\backslash sum\_\{j\backslash in\backslash mathbb\{N\}\}T\_j$ does not converge in the uniform operator topology.

Yet, since

$\backslash Vert\; T\_j\; T\_k^\backslash ast\backslash Vert=0$

and

$\backslash Vert\; T\_j^\backslash ast\; T\_k\backslash Vert=0$

for $j\backslash ne\; k$,

the Cotlar–Stein almost orthogonality lemma tells us that

:$T=\backslash sum\_\{j\backslash in\backslash mathbb\{N\}\}T\_j$

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

{{reflist|2}}

• {{citation|first=Mischa|last= Cotlar|title=A combinatorial inequality and its application to L² 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|first1=Anthony W.|last1=Knapp|first2=Elias|last2=Stein|title=Intertwining operators for semisimple Lie groups|journal=Ann. Math.|volume=93|year=1971|pages=489–579|doi=10.2307/1970887 |jstor=1970887 }}
• {{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}}

{{Functional analysis}}

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Category:Hilbert spaces

Category:Harmonic analysis

Category:Operator theory

Category:Inequalities (mathematics)

Category:Theorems in functional analysis

Category:Lemmas in mathematical analysis