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
[1] and allowed him to conclude that the
Hilbert transform is a
continuous linear operator in
L
2
{\displaystyle L^{2}}
without using the
Fourier transform . A more general version was proved by
Elias Stein .
[2]
Cotlar–Stein almost orthogonality lemma
Let
E
,
F
{\displaystyle E,\,F}
be two
Hilbert spaces . Consider a family of operators
T
j
{\displaystyle T_{j}}
,
j
≥
1
{\displaystyle j\geq 1}
, with each
T
j
{\displaystyle T_{j}}
a
bounded linear operator from
E
{\displaystyle E}
to
F
{\displaystyle F}
.
Denote
a
j
k
=
‖
T
j
T
k
∗
‖
,
b
j
k
=
‖
T
j
∗
T
k
‖
.
{\displaystyle a_{jk}=\Vert T_{j}T_{k}^{\ast }\Vert ,\qquad b_{jk}=\Vert T_{j}^{\ast }T_{k}\Vert .}
The family of operators
T
j
:
E
→
F
{\displaystyle T_{j}:\;E\to F}
,
j
≥
1
,
{\displaystyle j\geq 1,}
is almost orthogonal if
A
=
sup
j
∑
k
a
j
k
<
∞
,
B
=
sup
j
∑
k
b
j
k
<
∞
.
{\displaystyle A=\sup _{j}\sum _{k}{\sqrt {a_{jk}}}<\infty ,\qquad B=\sup _{j}\sum _{k}{\sqrt {b_{jk}}}<\infty .}
The Cotlar–Stein lemma states that if
T
j
{\displaystyle T_{j}}
are almost orthogonal, then the series
∑
j
T
j
{\displaystyle \sum _{j}T_{j}}
converges in the
strong operator topology , ↵and that
‖
∑
j
T
j
‖
≤
A
B
.
{\displaystyle \Vert \sum _{j}T_{j}\Vert \leq {\sqrt {AB}}.}
Proof
If T 1 , …, T n is a finite collection of bounded operators, then
[3]
∑
i
,
j
|
(
T
i
v
,
T
j
v
)
|
≤
(
max
i
∑
j
‖
T
i
∗
T
j
‖
1
2
)
(
max
i
∑
j
‖
T
i
T
j
∗
‖
1
2
)
‖
v
‖
2
.
{\displaystyle \displaystyle {\sum _{i,j}|(T_{i}v,T_{j}v)|\leq \left(\max _{i}\sum _{j}\|T_{i}^{*}T_{j}\|^{1 \over 2}\right)\left(\max _{i}\sum _{j}\|T_{i}T_{j}^{*}\|^{1 \over 2}\right)\|v\|^{2}.}}
So under the hypotheses of the lemma,
∑
i
,
j
|
(
T
i
v
,
T
j
v
)
|
≤
A
B
‖
v
‖
2
.
{\displaystyle \displaystyle {\sum _{i,j}|(T_{i}v,T_{j}v)|\leq AB\|v\|^{2}.}}
It follows that
‖
∑
i
=
1
n
T
i
v
‖
2
≤
A
B
‖
v
‖
2
,
{\displaystyle \displaystyle {\|\sum _{i=1}^{n}T_{i}v\|^{2}\leq AB\|v\|^{2},}}
and that
‖
∑
i
=
m
n
T
i
v
‖
2
≤
∑
i
,
j
≥
m
|
(
T
i
v
,
T
j
v
)
|
.
{\displaystyle \displaystyle {\|\sum _{i=m}^{n}T_{i}v\|^{2}\leq \sum _{i,j\geq m}|(T_{i}v,T_{j}v)|.}}
Hence, the partial sums
s
n
=
∑
i
=
1
n
T
i
v
{\displaystyle \displaystyle {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
=
∑
a
i
j
T
i
∗
T
j
{\displaystyle \displaystyle {R=\sum a_{ij}T_{i}^{*}T_{j}}}
with |a ij | ≤ 1 chosen so that
(
R
v
,
v
)
=
|
(
R
v
,
v
)
|
=
∑
|
(
T
i
v
,
T
j
v
)
|
.
{\displaystyle \displaystyle {(Rv,v)=|(Rv,v)|=\sum |(T_{i}v,T_{j}v)|.}}
Then
‖
R
‖
2
m
=
‖
(
R
∗
R
)
m
‖
≤
∑
‖
T
i
1
∗
T
i
2
T
i
3
∗
T
i
4
⋯
T
i
2
m
‖
≤
∑
(
‖
T
i
1
∗
‖
‖
T
i
1
∗
T
i
2
‖
‖
T
i
2
T
i
3
∗
‖
⋯
‖
T
i
2
m
−
1
∗
T
i
2
m
‖
‖
T
i
2
m
‖
)
1
2
.
{\displaystyle \displaystyle {\|R\|^{2m}=\|(R^{*}R)^{m}\|\leq \sum \|T_{i_{1}}^{*}T_{i_{2}}T_{i_{3}}^{*}T_{i_{4}}\cdots T_{i_{2m}}\|\leq \sum \left(\|T_{i_{1}}^{*}\|\|T_{i_{1}}^{*}T_{i_{2}}\|\|T_{i_{2}}T_{i_{3}}^{*}\|\cdots \|T_{i_{2m-1}}^{*}T_{i_{2m}}\|\|T_{i_{2m}}\|\right)^{1 \over 2}.}}
Hence
‖
R
‖
2
m
≤
n
⋅
max
‖
T
i
‖
(
max
i
∑
j
‖
T
i
∗
T
j
‖
1
2
)
2
m
(
max
i
∑
j
‖
T
i
T
j
∗
‖
1
2
)
2
m
−
1
.
{\displaystyle \displaystyle {\|R\|^{2m}\leq n\cdot \max \|T_{i}\|\left(\max _{i}\sum _{j}\|T_{i}^{*}T_{j}\|^{1 \over 2}\right)^{2m}\left(\max _{i}\sum _{j}\|T_{i}T_{j}^{*}\|^{1 \over 2}\right)^{2m-1}.}}
Taking 2m th roots and letting m tend to ∞,
‖
R
‖
≤
(
max
i
∑
j
‖
T
i
∗
T
j
‖
1
2
)
(
max
i
∑
j
‖
T
i
T
j
∗
‖
1
2
)
,
{\displaystyle \displaystyle {\|R\|\leq \left(\max _{i}\sum _{j}\|T_{i}^{*}T_{j}\|^{1 \over 2}\right)\left(\max _{i}\sum _{j}\|T_{i}T_{j}^{*}\|^{1 \over 2}\right),}}
which immediately implies the inequality.
Generalization
There is a generalization of the Cotlar–Stein lemma, with sums replaced by integrals.
[4]
[5] 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
=
sup
x
∫
X
‖
T
(
x
)
∗
T
(
y
)
‖
1
2
d
μ
(
y
)
,
B
=
sup
x
∫
X
‖
T
(
y
)
T
(
x
)
∗
‖
1
2
d
μ
(
y
)
,
{\displaystyle \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),}}
are finite, then the function T (x )v is integrable for each v in E with
‖
∫
X
T
(
x
)
v
d
μ
(
x
)
‖
≤
A
B
⋅
‖
v
‖
.
{\displaystyle \displaystyle {\|\int _{X}T(x)v\,d\mu (x)\|\leq {\sqrt {AB}}\cdot \|v\|.}}
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 infinite-dimensional matrices
T
=
1
0
0
⋮
0
1
0
⋮
0
0
1
⋮
⋯
⋯
⋯
⋱
{\displaystyle 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
T
1
=
1
0
0
⋮
0
0
0
⋮
0
0
0
⋮
⋯
⋯
⋯
⋱
,
T
2
=
0
0
0
⋮
0
1
0
⋮
0
0
0
⋮
⋯
⋯
⋯
⋱
,
T
3
=
0
0
0
⋮
0
0
0
⋮
0
0
1
⋮
⋯
⋯
⋯
⋱
,
…
.
{\displaystyle \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
‖
T
j
‖
=
1
{\displaystyle \Vert T_{j}\Vert =1}
for each
j
{\displaystyle j}
, hence the series
∑
j
∈
N
T
j
{\displaystyle \sum _{j\in \mathbb {N} }T_{j}}
does not converge in the
uniform operator topology .
Yet, since
‖
T
j
T
k
∗
‖
=
0
{\displaystyle \Vert T_{j}T_{k}^{\ast }\Vert =0}
and
‖
T
j
∗
T
k
‖
=
0
{\displaystyle \Vert T_{j}^{\ast }T_{k}\Vert =0}
for
j
≠
k
{\displaystyle j\neq k}
,
the Cotlar–Stein almost orthogonality lemma tells us that
T
=
∑
j
∈
N
T
j
{\displaystyle T=\sum _{j\in \mathbb {N} }T_{j}}
converges in the
strong operator topology and is bounded by 1.
Notes
References
Cotlar, Mischa (1955), "A combinatorial inequality and its application to L2 spaces", Math. Cuyana , 1 : 41–55
Hörmander, Lars (1994), Analysis of Partial Differential Operators III: Pseudodifferential Operators (2nd ed.), Springer-Verlag, pp. 165–166,
ISBN
978-3-540-49937-4
Knapp, Anthony W.; Stein, Elias (1971), "Intertwining operators for semisimple Lie groups", Ann. Math. , 93 : 489–579,
doi :
10.2307/1970887 ,
JSTOR
1970887
Stein, Elias (1993), Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals , Princeton University Press,
ISBN
0-691-03216-5
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