Cotlar–Stein lemma

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^[1] and allowed him to conclude that the Hilbert transform is a continuous linear operator in L² without using the Fourier transform. A more general version was proved by Elias Stein.^[2]

Cotlar–Stein almost orthogonality lemma

Let $E,\,F$ be two Hilbert spaces. Consider a family of operators T_j, $j\in\mathbb{Z}$, with each T_j a continuous linear operator from E to F.

Denote

$a_{jk}=\Vert T_j T_k^\ast\Vert_{F\to F}, \qquad b_{jk}=\Vert T_j^\ast T_k\Vert_{E\to E}.$

The family of operators $T_j:\;E\to F$, $j\in\mathbb{Z},$ is almost orthogonal if

$A=\max_{j}\sum_{k}\sqrt{a_{jk}}<\infty, \qquad B=\max_{j}\sum_{k}\sqrt{b_{jk}}<\infty.$

The Cotlar–Stein lemma states that if T_j are almost orthogonal, then the series $\sum_{j\in\mathbb{Z}}T_j$ converges in the strong operator topology, and that

$\Vert \sum_{j\in\mathbb{Z}}T_j\Vert_{E\to F}\le\sqrt{AB}.$

Example

Here is an example of an orthogonal family of operators. Consider the inifite-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 $\Vert T_j\Vert=1$ 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

1. ^ Mischa Cotlar, A combinatorial inequality and its application to L² spaces, Math. Cuyana 1 (1955), 41–55
2. ^ Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993. ISBN 0-691-03216-5