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]
Statement of the lemma [ edit ]
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
‖
∑
j
T
j
‖
≤
A
B
.
{\displaystyle \Vert \sum _{j}T_{j}\Vert \leq {\sqrt {AB}}.}
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 [ edit ]
The Cotlar-Stein lemma has been generalized, with sums being 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 proven by replacing sums with integrals in the previous proof, or by utilizing Riemann sums to approximate the integrals.
Example [ edit ]
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.
References [ edit ]
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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