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 ] 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 [ 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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