COTLAR–STEIN LEMMA

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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
and allowed him to conclude that the Hilbert transform
is a continuous linear operator in $L^2$
without using the Fourier transform.

Contents

Cotlar–Stein almost orthogonality lemma

Example

References

Cotlar–Stein almost orthogonality lemma

Let $E,,F$ be two Hilbert spaces.
Consider a family of operators
$T\_j$, $jinmathbb\{Z\}$,
with each $T\_j$
a continuous linear operator from $E$ to $F$.
Denote
:
qquad b_{jk}=Vert T_j^st T_kVert_{E o E}.
The family of operators
$T\_j:;E\; o\; F$, $jinmathbb\{Z\},$
is ''almost orthogonal'' if
:
The Cotlar-Stein lemma states that if $T\_j$
are almost orthogonal,
then the series
$sum\_\{jinmathbb\{Z\}\}T\_j$
converges in the strong operator topology,
and that
:$Vert\; sum\_\{jinmathbb\{Z\}\}T\_jVert\_\{E\; o\; F\}lesqrt\{AB\}.$

Example

Here is an example of an ''orthogonal'' family of operators. Consider the inifite-dimensional matrices
:$$
T=left[
egin{array}{cccc}
1&0&0&dots\0&1&0&dots\0&0&1&dots\cdots&cdots&cdots&ddotsend{array}
ight]

and also
:$$
qquad
T_1=left[
egin{array}{cccc}
1&0&0&dots\0&0&0&dots\0&0&0&dots\cdots&cdots&cdots&ddotsend{array}
ight],
qquad
T_2=left[
egin{array}{cccc}
0&0&0&dots\0&1&0&dots\0&0&0&dots\cdots&cdots&cdots&ddotsend{array}
ight],
qquad
T_3=left[
egin{array}{cccc}
0&0&0&dots\0&0&0&dots\0&0&1&dots\cdots&cdots&cdots&ddotsend{array}
ight],
qquad
dots.

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

and

for $j$
e k,
the Cotlar-Stein almost orthogonality lemma tells us that
:$T=sum\_\{jinmathbb\{N\}\}T\_j$
converges in the strong operator topology and is bounded by 1.

References

★ Mischa Cotlar, ''A combinatorial inequality and its application to $L^2$ spaces'', Math. Cuyana 1 (1955), 41-55

★ Elias Stein, ''Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals''. Princeton University Press, 1993. ISBN 0-691-03216-5