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 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 $T_j$, $j\ge 1$, with each $T_j$ a bounded linear operator from $E$ to $F$.

Denote

$a_{jk}=\Vert T_j{T}_k^{\ast }\Vert ,b_{jk}=\Vert T_j^{\ast }{T}_k\Vert .$

The family of operators $T_j:E\to F$, $j\ge 1,$ is almost orthogonal if

$A=\underset{j}{\sup }\sum _k\sqrt{a_{jk}}<\mathrm{\infty },B=\underset{j}{\sup }\sum _k\sqrt{b_{jk}}<\mathrm{\infty }.$

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

$\Vert \sum _j{T}_j\Vert \le \sqrt{AB}.$

Proof

If $T_1,\dots ,T_n$ is a finite collection of bounded operators, then

$\sum _{i,j}|(T_{i}v,T_{j}v)|\le (\underset{i}{\max }\sum _j\Vert T_i^{*}{T}_j{\Vert }^{\frac{1}{2}})(\underset{i}{\max }\sum _j\Vert T_i{T}_j^{*}{\Vert }^{\frac{1}{2}})\Vert v{\Vert }^2.$

So under the hypotheses of the lemma,

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

It follows that

$\Vert \sum _{i=1}^n{T}_{i}v{\Vert }^2\le AB\Vert v{\Vert }^2,$

and that

$\Vert \sum _{i=m}^n{T}_{i}v{\Vert }^2\le \sum _{i,j\ge m}|(T_{i}v,T_{j}v)|.$

Hence, the partial sums

$s_n=\sum _{i=1}^n{T}_{i}v$

form a Cauchy sequence.

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

To prove the inequality above set

$R=\sum a_{ij}{T}_i^{*}{T}_j$

with |a_ij| ≤ 1 chosen so that

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

Then

$\Vert R{\Vert }^{2m}=\Vert (R^{*}R{)}^m\Vert \le \sum \Vert T_{i_1}^{*}{T}_{i_2}{T}_{i_3}^{*}{T}_{i_4}\cdots T_{i_{4m}}\Vert \le \sum {(\Vert T_{i_1}^{*}\Vert \Vert T_{i_1}^{*}{T}_{i_2}\Vert \Vert T_{i_2}{T}_{i_3}^{*}\Vert \cdots \Vert T_{i_{4m-1}}^{*}{T}_{i_{4m}}\Vert \Vert T_{i_{4m}}\Vert )}^{\frac{1}{2}}.$

Hence

$\Vert R{\Vert }^{2m}\le n\cdot \max \Vert T_i\Vert {(\underset{i}{\max }\sum _j\Vert T_i^{*}{T}_j{\Vert }^{\frac{1}{2}})}^{2m}{(\underset{i}{\max }\sum _j\Vert T_i{T}_j^{*}{\Vert }^{\frac{1}{2}})}^{2m-1}.$

Taking 2mth roots and letting m tend to ∞,

$\Vert R\Vert \le (\underset{i}{\max }\sum _j\Vert T_i^{*}{T}_j{\Vert }^{\frac{1}{2}})(\underset{i}{\max }\sum _j\Vert T_i{T}_j^{*}{\Vert }^{\frac{1}{2}}),$

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

$A=\underset{x}{\sup }\underset{X}{\int }\Vert T(x{)}^{*}T(y){\Vert }^{\frac{1}{2}}d\mu (y),B=\underset{x}{\sup }\underset{X}{\int }\Vert T(y)T(x{)}^{*}{\Vert }^{\frac{1}{2}}d\mu (y),$

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

$\Vert \underset{X}{\int }T(x)vd\mu (x)\Vert \le \sqrt{AB}\cdot \Vert v\Vert .$

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.

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

and also

$T_1=\left[\begin{array}{cccc}1& 0& 0& ⋮\\ 0& 0& 0& ⋮\\ 0& 0& 0& ⋮\\ \cdots & \cdots & \cdots & \ddots \end{array}\right],T_2=\left[\begin{array}{cccc}0& 0& 0& ⋮\\ 0& 1& 0& ⋮\\ 0& 0& 0& ⋮\\ \cdots & \cdots & \cdots & \ddots \end{array}\right],T_3=\left[\begin{array}{cccc}0& 0& 0& ⋮\\ 0& 0& 0& ⋮\\ 0& 0& 1& ⋮\\ \cdots & \cdots & \cdots & \ddots \end{array}\right],\dots .$

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

Yet, since $\Vert T_j{T}_k^{\ast }\Vert =0$ and $\Vert T_j^{\ast }{T}_k\Vert =0$ for $j\ne k$, the Cotlar–Stein almost orthogonality lemma tells us that

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

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

References

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